Derivative Calculator

Use this Derivative Calculator to find the derivative of a function, calculate first, second, third or fourth derivatives, evaluate a derivative at a specific value of x, find the tangent line to a curve, and compare a symbolic derivative with a numerical approximation. Enter a function such as x^3 + 2x, sin(x^2), ln(x), e^x or a quotient and the calculator will work through the differentiation rules automatically.

Derivatives are one of the central ideas in calculus because they measure how a function changes. They can describe the slope of a curve, instantaneous speed, acceleration, growth rates, optimization problems and many other changing quantities. This calculator is designed to provide both an answer and useful calculation details so you can check calculus homework, study differentiation rules or solve practical derivative problems.

How to Use the Derivative Calculator

The calculator has four modes because derivative problems can ask for several different kinds of answers. The first mode finds a symbolic derivative. The second evaluates that derivative at a particular value of x. The third uses the derivative to calculate the equation of a tangent line, and the fourth checks a derivative numerically using values immediately to the left and right of a point.

For a standard derivative problem, choose Find Derivative. Type your function into the f(x) field, select the order of derivative you need, and press Calculate Derivative. The calculator accepts ordinary polynomial expressions such as x^3 + 4x^2 – 7x + 2 as well as products, fractions, trigonometric functions, logarithms, exponentials and square roots.

You can write multiplication explicitly as 2*x, but the calculator also recognizes several common forms such as 2x, 3sin(x) and 2(x+1). Exponents are entered with the ^ symbol, so x squared is written x^2 and x to the fifth power is x^5. Parentheses are important whenever one complete expression is being raised to a power, divided by another expression or placed inside a function.

For trigonometric functions, enter sin(x), cos(x), tan(x), sec(x), csc(x) or cot(x). The calculator treats the variable as a real-number radian input, which is the standard convention when differentiating trigonometric functions in calculus. It also supports asin(x), acos(x), atan(x), ln(x), log(x), exp(x) and sqrt(x).

If you need to simplify, rearrange or solve an equation before differentiating it, the Algebra Calculator can help with linear equations, quadratics, systems, inequalities and expression work.

What Is a Derivative?

A derivative measures the instantaneous rate at which one quantity changes with respect to another. If y = f(x), the derivative f′(x) describes how rapidly y is changing as x changes. Geometrically, the derivative gives the slope of the tangent line to the graph of the function at a point where the derivative exists.

Consider the function f(x) = x². Its derivative is f′(x) = 2x. At x = 1 the derivative equals 2, while at x = 5 it equals 10. The curve is therefore getting steeper as x increases. The derivative is not one fixed slope for the entire parabola; it is itself a function that tells you the slope at each x-value.

This idea connects calculus to real-world rates. If a position function describes where a moving object is located over time, its first derivative describes instantaneous velocity. Differentiating velocity gives acceleration. If a function describes the amount of money earned from selling a changing number of units, its derivative can describe marginal revenue. Similar interpretations appear in physics, engineering, economics, biology and optimization.

The derivative can also be understood as the limiting value of secant-line slopes. A secant line passes through two points on a curve. As the second point moves closer to the first, the slope of the secant line approaches the slope of the tangent line. That limiting slope is the derivative.

Derivative Formula

The formal definition of the derivative at x uses a limit. The most common version is:

The expression in brackets is a difference quotient. The numerator measures how much the function changes when x increases by h, and the denominator measures the change in x. Dividing gives an average rate of change over the small interval. Taking the limit as h approaches zero turns that average rate into an instantaneous rate of change.

Although the limit definition is fundamental, using it from scratch for every derivative would be slow. Calculus therefore develops a set of differentiation rules that produce the same result much more efficiently. The power rule, product rule, quotient rule and chain rule are among the most important.

For a formal treatment of these rules, the OpenStax Calculus differentiation rules provide detailed explanations of the constant, power, product and quotient rules.

Basic Derivative Rules

Differentiation becomes much easier once the basic rules are recognized. A complicated function can often be broken into smaller pieces. Each piece is differentiated with the rule that matches its structure, and the results are then combined.

A constant such as 7 does not change as x changes, so its derivative is zero. The derivative of x is 1. Constant multiples remain attached to the derivative, so the derivative of 5x³ is 5 times the derivative of x³. Sums and differences are differentiated term by term.

When you are checking ordinary arithmetic inside a derivative calculation, the Simple Math Calculator can be useful for addition, subtraction, multiplication and division without mixing that arithmetic into the calculus step itself.

Power Rule

The power rule is one of the fastest and most frequently used derivative rules. If a variable is raised to a constant power n, multiply by the exponent and then reduce the exponent by one.

For example, the derivative of x⁵ is 5x⁴. The derivative of 7x³ is 21x² because the constant 7 remains attached while the derivative of x³ is 3x². A polynomial can therefore be differentiated one term at a time.

Suppose f(x) = 4x⁴ − 3x² + 6x − 10. Differentiate each term separately. The derivative of 4x⁴ is 16x³. The derivative of −3x² is −6x. The derivative of 6x is 6, and the derivative of −10 is zero. The final derivative is f′(x) = 16x³ − 6x + 6.

The rule also works for negative and fractional powers when the function is defined. For example, x⁻² differentiates to −2x⁻³. Since x⁻² is the same as 1/x², this is equivalent to −2/x³.

Product Rule

When two functions are multiplied together, you generally cannot differentiate each one and simply multiply the two derivatives. Instead, use the product rule.

The rule says to differentiate the first factor while leaving the second unchanged, then add the first factor unchanged multiplied by the derivative of the second.

For example, suppose f(x) = x² sin(x). Let u = x² and v = sin(x). The derivative of u is 2x, and the derivative of v is cos(x). The product rule gives f′(x) = 2x sin(x) + x² cos(x).

It is worth recognizing when an expression is truly a product and when ordinary algebra can simplify it first. For example, x² multiplied by x³ can be rewritten as x⁵ before differentiating. In that case, the power rule is quicker. The Factor Calculator can help with factors and related algebra when you are preparing expressions for calculus work.

Quotient Rule

The quotient rule differentiates one function divided by another. Like the product rule, it is not correct to differentiate the numerator and denominator separately and then divide the two derivatives.

Suppose f(x) = x² / (x + 1). Let the numerator be u = x² and the denominator be v = x + 1. Then u′ = 2x and v′ = 1. Substitution into the quotient rule gives [2x(x + 1) − x²] / (x + 1)². That expression can then be simplified if desired.

Whenever a denominator can equal zero, remember that the original function is not defined at those x-values. Differentiation does not remove that original domain restriction. If you are working heavily with rational expressions or need to simplify a numerical fraction separately, the Fraction Calculator can help with fraction arithmetic and simplification.

Chain Rule

The chain rule is used for composite functions, where one function appears inside another. It is one of the most important ideas in differentiation because many expressions that appear complicated are really combinations of simple functions nested together.

Take sin(x²) as an example. The outside function is sine, and the inside function is x². Differentiate the outside while leaving the inside in place: cos(x²). Then multiply by the derivative of the inside, which is 2x. The result is 2x cos(x²).

For (3x + 1)⁵, the outside operation is raising something to the fifth power. Differentiating the outside gives 5(3x + 1)⁴. The inside function is 3x + 1, whose derivative is 3. Multiplying gives 15(3x + 1)⁴.

The chain rule can be combined with the product and quotient rules. A function such as x² sin(x³) requires a product rule because two functions are multiplied, and differentiating sin(x³) inside that product also requires the chain rule. Complex derivative problems often involve several rules layered together rather than one isolated formula.

Khan Academy’s chain rule and advanced derivatives material provides additional examples of composite functions, implicit differentiation, second derivatives and combinations of several differentiation rules.

Trigonometric Derivatives

Trigonometric functions have standard derivative formulas. In calculus these formulas assume angles are measured in radians. The derivative of sin(x) is cos(x), while the derivative of cos(x) is −sin(x). The derivative of tan(x) is sec²(x).

The reciprocal trigonometric functions also have derivative rules. The derivative of sec(x) is sec(x)tan(x). The derivative of csc(x) is −csc(x)cot(x), and the derivative of cot(x) is −csc²(x).

When the input contains more than x alone, the chain rule also applies. The derivative of sin(3x) is 3cos(3x), not merely cos(3x). The derivative of cos(x²) is −2x sin(x²). The inner derivative is what distinguishes these from the basic trig formulas.

If you need to evaluate sine, cosine, tangent, logarithms or powers numerically, the Scientific Calculator provides a separate numeric tool for those operations.

Inverse Trigonometric Derivatives

Inverse trigonometric functions have their own derivatives. The derivative of arcsin(x), written asin(x) in this calculator, is 1 divided by sqrt(1 − x²). The derivative of arccos(x) is the negative of that expression. The derivative of arctan(x) is 1 divided by 1 + x².

The domain of these functions matters. For example, asin(x) and acos(x) are real-valued only for x between −1 and 1. Their derivatives also have restrictions at the endpoints because sqrt(1 − x²) becomes zero at x = ±1.

Composite versions require the chain rule. For asin(2x), differentiate the inverse sine formula and multiply by 2, the derivative of the inner function.

Logarithmic and Exponential Derivatives

The natural exponential function is unusual because its derivative is the function itself. If f(x) = e^x, then f′(x) = e^x. The calculator uses exp(x) to represent e raised to a function when convenient, so exp(2x) means e^(2x).

With a composite exponent, use the chain rule. The derivative of e^(3x) is 3e^(3x). More generally, the derivative of e^(g(x)) is e^(g(x)) multiplied by g′(x).

The natural logarithm has derivative 1/x when x is positive. For ln(g(x)), the chain rule produces g′(x)/g(x). For example, the derivative of ln(x² + 1) is 2x/(x² + 1).

This calculator interprets log(x) as a base-10 logarithm. Its derivative is 1/[x ln(10)]. That constant ln(10) appears because changing the base of a logarithm changes its scaling.

Derivative of a Square Root

A square root can be rewritten as a fractional power. Since sqrt(x) = x^(1/2), the power rule gives a derivative of 1/[2sqrt(x)].

For sqrt(x² + 4), the chain rule is also needed. Differentiate the outside square root and multiply by the derivative of x² + 4. The resulting derivative is x/sqrt(x² + 4).

When using square roots, remember the original real-valued function may require its input to be nonnegative. A symbolic derivative formula should always be interpreted together with the domain of the original function.

Second Derivative Calculator

The second derivative is found by differentiating the first derivative. If f′(x) describes the rate at which the original function changes, then f″(x) describes how that rate itself changes.

For f(x) = x³, the first derivative is 3x². Differentiate again and the second derivative is 6x. A third derivative gives 6, and a fourth derivative gives zero.

Second derivatives are especially useful for studying concavity. When f″(x) is positive over an interval, the graph is typically concave upward there. When f″(x) is negative, the graph is typically concave downward. Changes in concavity are related to possible inflection points.

In motion problems, if position is a function of time, the first derivative is velocity and the second derivative is acceleration. Higher derivatives can describe how acceleration itself changes.

The calculator allows first, second, third and fourth derivatives. Higher orders can become algebraically large for complicated functions, which is why the tool limits the order rather than generating an unlimited expression that becomes difficult to read or use.

Derivative at a Point

A symbolic derivative gives a formula for the slope at any allowable value of x. Sometimes a problem instead asks for the slope at one specific point. In that case, first differentiate the function and then substitute the requested x-value into the derivative.

For f(x) = x² + 3x + 1, the derivative is f′(x) = 2x + 3. At x = 2, substitute 2 into the derivative: 2(2) + 3 = 7. The slope of the function at x = 2 is therefore 7.

The calculator’s Derivative at a Point mode performs both steps. It also evaluates the original function so you can see the y-coordinate of the point. In the example above, f(2) = 11, so the point on the curve is (2, 11) and the tangent slope there is 7.

Tangent Line Calculator

Once you know a derivative at a point, you can construct the tangent line. The tangent line is the straight line that has the same instantaneous slope as the curve at that point.

Point-slope form is particularly convenient:

Here m is the derivative evaluated at the point, x₁ is the chosen x-coordinate, and y₁ is the original function evaluated at x₁.

For f(x) = x² at x = 3, the derivative is 2x. The slope at x = 3 is therefore 6. The original function gives f(3) = 9, so the point is (3, 9). The tangent equation is y − 9 = 6(x − 3), which simplifies to y = 6x − 9.

The Tangent Line mode calculates the function value, derivative value, point-slope equation and expanded line equation automatically. This is particularly useful for calculus problems involving linear approximation or local behavior of a curve.

Numerical Derivative Calculator

Symbolic differentiation applies rules to produce an exact algebraic expression. Numerical differentiation takes a different approach: it estimates the slope using function values very close to the point of interest.

This calculator uses a central difference approximation:

The value h is a small step. The calculation evaluates the function slightly to the right and slightly to the left of x, then uses those two points to estimate the local slope. A reasonably small h often produces a result close to the symbolic derivative.

Making h smaller does not improve the answer forever. Computers store numbers with limited precision, so extremely tiny steps can introduce floating-point roundoff. Numerical differentiation is therefore a balance between using a step small enough to approximate the tangent and large enough to avoid excessive numerical error.

The Numerical Check mode displays both the numerical estimate and the symbolic derivative evaluated at the same x-value. Their absolute difference provides a quick way to see how closely the approximation agrees with the symbolic result.

Derivative Examples

Example 1: Derivative of x³ + 2x

Differentiate each term separately. The derivative of x³ is 3x², while the derivative of 2x is 2. Therefore:

f′(x) = 3x² + 2

This is a straightforward combination of the power rule, constant-multiple rule and sum rule.

Example 2: Derivative of sin(x²)

This is a composite function. The outside function is sine and the inside function is x². Differentiate sine to get cosine, keep x² inside, and multiply by the derivative of x²:

f′(x) = 2x cos(x²)

This is a classic chain-rule example.

Example 3: Derivative of (x² + 1)/(x − 1)

This requires the quotient rule. Let u = x² + 1 and v = x − 1. Then u′ = 2x and v′ = 1. Substitute into the quotient formula:

f′(x) = [2x(x − 1) − (x² + 1)]/(x − 1)²

The derivative may be expanded or simplified further, but this form already shows the quotient-rule structure clearly.

Example 4: Derivative of e^(2x)

The derivative of e to a function is the same exponential multiplied by the derivative of the exponent. Since the derivative of 2x is 2:

f′(x) = 2e^(2x)

Example 5: Derivative of ln(x)

The natural logarithm has one of the standard calculus derivatives:

f′(x) = 1/x

For a composite logarithm such as ln(x² + 1), the result becomes 2x/(x² + 1) after applying the chain rule.

Example 6: Derivative of sqrt(x)

Rewrite sqrt(x) as x^(1/2), then apply the power rule:

f′(x) = 1/[2sqrt(x)]

Example 7: Derivative of x^x

The expression x^x is different from an ordinary power such as x³ because both the base and exponent depend on x. Logarithmic differentiation or the general power rule gives:

f′(x) = x^x[ln(x) + 1]

This is a useful example of why derivative problems cannot always be solved with the ordinary power rule alone.

Derivative of a Polynomial

Polynomials are among the easiest functions to differentiate because every term can be handled independently with the power rule. There is no need for the product or quotient rule unless the polynomial has deliberately been left in a multiplied or divided form.

Consider f(x) = 6x⁵ − 2x³ + 9x² − 4x + 11. Differentiate term by term. The first derivative is 30x⁴ − 6x² + 18x − 4. The constant 11 disappears because its derivative is zero.

If you then request the second derivative, differentiate that result again. The second derivative is 120x³ − 12x + 18. Repeated differentiation eventually reduces every nonconstant polynomial to zero.

Derivative of a Rational Function

A rational function is a quotient of polynomials, such as (x² + 3)/(x − 4). These functions often require the quotient rule unless algebraic cancellation can simplify the expression first.

Domain restrictions are especially important. In the example above, x cannot equal 4 because that would make the denominator zero. Even if an algebraic simplification appears to cancel a factor, the original excluded value may still matter when describing the original function.

Ratios and quotients are related but not identical concepts. If you are working on a proportion problem rather than differentiating a function, use the Ratio Calculator for direct ratio and proportion calculations.

Derivative Notation

Calculus uses several notations for derivatives. If y = f(x), the first derivative may be written f′(x), y′ or dy/dx. These notations express the same basic idea but emphasize different viewpoints.

Leibniz notation dy/dx emphasizes one variable changing with respect to another. Prime notation f′(x) is compact and convenient when repeatedly differentiating the same function. The second derivative can be written f″(x) or d²y/dx².

Higher derivatives continue the pattern. The third derivative may be written f‴(x), while higher orders are often written with a number in parentheses, such as f⁽⁴⁾(x) for the fourth derivative.

Differentiable vs. Continuous

A function must be sufficiently smooth at a point for an ordinary derivative to exist there. A differentiable function is continuous at that point, but the reverse is not always true. A function can be continuous and still have a sharp corner or cusp where there is no single tangent slope.

The absolute-value function f(x) = |x| is a classic example. It is continuous at x = 0, but the slope approaching from the left is −1 while the slope approaching from the right is +1. Because those one-sided slopes disagree, the derivative at zero does not exist.

Vertical tangents, discontinuities and domain boundaries can also create points where a derivative is undefined. A symbolic formula should therefore not be interpreted as proof that a derivative exists at every real number.

Critical Points

A critical point generally occurs where the derivative equals zero or where the derivative is undefined while the original function is defined. Critical points are important because local maxima and minima can occur there.

For f(x) = x² − 4x + 7, the derivative is 2x − 4. Setting the derivative equal to zero gives x = 2. Evaluating the original function gives f(2) = 3. Because the parabola opens upward, this point is the minimum of the function.

Finding critical points usually requires two separate skills: differentiation and solving the resulting equation. The Derivative Calculator can produce f′(x), while the Algebra Calculator can help solve many equations created by setting that derivative equal to zero.

Increasing and Decreasing Functions

The sign of the first derivative gives information about the direction a function is moving. If f′(x) is positive across an interval, the function is increasing there. If f′(x) is negative, the function is decreasing.

This does not mean the derivative itself must be increasing. A function can have a positive derivative that is becoming smaller, meaning the function still rises but at a slowing rate.

Sign analysis is often performed by finding critical points, splitting the number line into intervals and testing the sign of f′(x) in each interval.

Concavity and the Second Derivative

The second derivative helps describe how the slope changes. If f″(x) is positive, slopes are generally increasing and the graph is concave upward. If f″(x) is negative, slopes are generally decreasing and the graph is concave downward.

An inflection point may occur where concavity changes. Merely finding a point where f″(x) = 0 is not enough by itself; the sign of the second derivative should change across the point for a genuine change in concavity.

The second derivative also appears in optimization tests. If f′(c) = 0 and f″(c) is positive, c is commonly a local minimum. If f″(c) is negative, c is commonly a local maximum, assuming the relevant derivative conditions are satisfied.

Applications of Derivatives

Derivatives are useful because many real-world questions involve change rather than a fixed quantity. A function can model a quantity, and its derivative can reveal how that quantity responds as the input changes.

In motion, position differentiated with respect to time gives velocity. Velocity differentiated again gives acceleration. In economics, derivatives are used in marginal cost, marginal revenue and optimization. In engineering, they appear in rates of heat transfer, electrical systems, mechanics, fluid flow and control systems.

Biology and population models use derivatives to describe growth rates. Chemistry uses rates of reaction. Finance uses changing rates in models of prices, risk and growth. Computer graphics and numerical analysis use derivatives for optimization and curve behavior.

Not every real-world model is differentiable everywhere, and a derivative is only as meaningful as the function being modeled. The mathematics gives the rate of change implied by the model; interpreting that rate still requires understanding the units and assumptions behind the original function.

Average Rate of Change vs. Instantaneous Rate of Change

Average rate of change uses two different input values. It measures the change in the function divided by the change in x over an interval. Geometrically, it is the slope of a secant line connecting two points on the graph.

Instantaneous rate of change is what the derivative measures. Instead of keeping two separate x-values, calculus takes a limit as their separation approaches zero. The secant slope approaches the tangent slope.

This distinction is similar to the difference between average speed over an entire trip and the speed shown by a speedometer at one particular instant. Both are rates, but they answer different questions.

Derivative vs. Integral

Derivatives and integrals are the two central operations of calculus. A derivative describes local change, while an integral accumulates quantities over an interval. Under the Fundamental Theorem of Calculus, differentiation and integration are closely related inverse processes under appropriate conditions.

If velocity is the derivative of position, integrating velocity over time can recover a change in position. If a function describes a rate of flow, integrating that rate can calculate the total amount accumulated.

This calculator focuses only on differentiation. Keeping derivative and integral tools separate makes each calculator easier to use and allows the explanations to match the specific operation being performed.

Common Derivative Mistakes

Many differentiation errors come from applying a correct rule to the wrong structure. Recognizing the outermost operation in a function is often the most important first step.

• Forgetting to reduce the exponent: The derivative of x⁵ is 5x⁴, not 5x⁵.
• Multiplying derivatives in a product: The derivative of fg is not simply f′g′.
• Dividing derivatives in a quotient: The derivative of f/g is not f′/g′.
• Forgetting the chain rule: The derivative of sin(x²) requires multiplication by 2x.
• Dropping a negative sign: The derivative of cos(x) is −sin(x).
• Ignoring domain restrictions: A formula may be undefined at particular x-values.
• Using degrees in derivative identities: Standard trig derivative formulas assume radians.
• Expanding too early: Sometimes leaving factors grouped makes the derivative easier to understand and less prone to arithmetic errors.

Why Two Correct Derivatives Can Look Different

Symbolic mathematics frequently produces expressions that are algebraically equivalent but look different. One derivative might be factored while another is expanded. A quotient may be written with separate terms or one common denominator. Trigonometric identities can also transform one correct form into another.

For example, 2x(x − 1) − (x² + 1) can be expanded to x² − 2x − 1. Both numerators represent the same polynomial after simplification. A calculator may therefore produce a result that differs visually from a textbook answer without being mathematically different.

When comparing results, simplify both expressions or test several valid x-values. If the difference between the two expressions simplifies to zero over their shared domain, they are equivalent.

Why Use a Derivative Calculator?

A derivative calculator can save time when checking long calculations, especially when several rules must be combined. It can also help students identify which rule applies to a function and verify work completed by hand.

The calculator is particularly useful for higher derivatives because small algebra mistakes in the first derivative can carry into every later step. Comparing your hand calculation with an independent symbolic result can reveal where an error first occurred.

It is still useful to learn the rules rather than treating a calculator as a substitute for calculus. Knowing the power, product, quotient and chain rules allows you to recognize whether an output makes sense, understand equivalent forms and solve problems when a calculator is unavailable.

Derivative Calculator Frequently Asked Questions

Final Thoughts

A derivative turns a function into a description of how that function changes. For a graph, it gives tangent slope. For motion, it can describe velocity or acceleration. For optimization, it helps locate points where a quantity stops increasing and begins decreasing or where the opposite occurs.

Use Find Derivative when you need a symbolic first or higher-order derivative. Use Derivative at a Point when you need the slope at one x-value. Use Tangent Line when you need the equation of the line that touches the curve with the same local slope. Numerical Check provides an independent approximation using nearby function values.

The most important differentiation skills remain recognizing the structure of the function and choosing the appropriate rule. Polynomials usually rely on the power rule, products require the product rule, quotients require the quotient rule, and nested functions require the chain rule. Many advanced expressions combine several of these ideas in a single problem.