Derivative Calculator with Steps
Differentiate polynomial, rational, trigonometric, exponential, logarithmic, root and absolute-value functions. Find a first, second or third derivative, evaluate it at a point and review the rules used.
Last Updated: July 26, 2026Online Differentiation Calculator
Enter f(x), choose the derivative order and optionally enter an x-value. Use radians for all trigonometric functions.
Symbolic result
f′(x) = 3·x^2 - 4
First derivative found. At x = 2, the instantaneous rate of change is 8.
Differentiation steps
- Read f(x) = x^3 - 4·x + 1 and differentiate with respect to x.
- Apply the sum, constant multiple and power rules.
- Simplify the result to f′(x) = 3·x^2 - 4.
- Substitute x = 2 to obtain f′(2) = 8.
How to Use This Derivative Calculator
- Enter a real-valued function of x. Use parentheses around every function argument, such as sin(x), ln(x) or sqrt(x + 1).
- Choose the first, second or third derivative. The first derivative measures rate of change, while higher derivatives describe how earlier rates change.
- Enter an optional x-value to calculate f(x) and the selected derivative at a point. The point field accepts constants and constant expressions such as pi/2.
- Select the display precision for numerical results, then choose Differentiate Function.
- Review the symbolic derivative, point values, applicable line equations and rule-based steps.
The default example uses f(x) = x3 - 4x + 1. Its first derivative is 3x2 - 4. At x = 2, the derivative equals 8, so the curve has tangent slope 8 at the point (2, 1).
What Is a Derivative?
A derivative measures the instantaneous rate at which one quantity changes relative to another. Geometrically, f′(a) is the slope of the tangent line to y = f(x) at x = a, when that derivative exists. In an applied problem, it may describe velocity, acceleration, marginal cost, growth or sensitivity.
The formal definition uses a limit of average rates of change over shorter intervals. The derivative exists only when this difference quotient approaches one finite value.
Differentiation rules avoid rebuilding this limit for every familiar function. The calculator parses the expression into a mathematical tree, applies the appropriate rules to each part and simplifies the resulting tree. It reports a symbolic formula rather than estimating the derivative from two nearby decimal points.
Core Differentiation Rules
Most introductory derivatives combine a small set of rules. In the table, u and v are differentiable functions of x, while c and n are constants.
| Rule | Function | Derivative | Use |
|---|---|---|---|
| Constant | c | 0 | A fixed value does not change. |
| Power | un | n·un-1·u′ | Powers and roots with a constant exponent |
| Sum or difference | u ± v | u′ ± v′ | Differentiate terms separately. |
| Product | u·v | u′·v + u·v′ | Two multiplied functions |
| Quotient | u/v | (u′·v - u·v′)/v2 | One function divided by another |
| Chain | F(u) | F′(u)·u′ | A function nested inside another function |
The derivative of a product is not the product of the derivatives. The derivative of a quotient is not the quotient of the derivatives. Both structures need their complete rules. A nested expression such as sin(x2) also needs the chain rule because the sine function acts on x2, not directly on x.
Worked Example: Differentiate a Polynomial
Differentiate f(x) = x3 - 4x + 1. Apply the sum and difference rules so each term is handled separately. Then apply the power rule and constant rule.
At x = 2, f′(2) = 3(22) - 4 = 8. The original function value is f(2) = 8 - 8 + 1 = 1. Therefore, the point is (2, 1) and the tangent line has slope 8.
The normal line is perpendicular to the tangent. Its slope is the negative reciprocal, -1/8, so its equation is y - 1 = -(1/8)(x - 2).
Product, Quotient and Chain Rule Examples
Product rule example
For f(x) = x sin(x), let u = x and v = sin(x). The derivatives are u′ = 1 and v′ = cos(x). The product rule gives:
Quotient rule example
For f(x) = (x2 + 1)/(x - 1), differentiate the numerator and denominator, preserve their positions and square the original denominator.
The original function and its derivative are undefined at x = 1. Algebraic simplification must not erase this domain restriction.
Chain rule example
For f(x) = sin(x2), the outer function is sine and the inner function is x2. Differentiate the outer function, keep the inner expression, then multiply by the inner derivative.
Trigonometric, Exponential and Logarithmic Derivatives
Trigonometric derivative formulas assume radian measure. Using degrees without the conversion factor changes the derivative, so this tool uses radians consistently.
| Function | Derivative with inner function u | Domain note |
|---|---|---|
| sin(u) | cos(u)·u′ | All real u |
| cos(u) | -sin(u)·u′ | All real u |
| tan(u) | sec2(u)·u′ | Undefined where cos(u) = 0 |
| asin(u) | u′/sqrt(1 - u2) | Real derivative for |u| < 1 |
| exp(u) | exp(u)·u′ | All real u |
| ln(u) | u′/u | Real input requires u > 0 |
| log(u) | u′/[u ln(10)] | log means base 10 and u > 0 |
| sqrt(u) | u′/[2 sqrt(u)] | Interior formula requires u > 0 |
| abs(u) | sign(u)·u′ | Undefined where u = 0 |
The parser also supports sec, csc, cot, sinh, cosh and tanh. Function arguments need parentheses. Type cos(x), not cos x. Constants pi and e are available, and ordinary implicit multiplication works in forms such as 2x, 3(x + 1) and 2sin(x).
Variable Exponents and Logarithmic Differentiation
A power such as xx has both a changing base and a changing exponent. The ordinary constant-exponent power rule is not enough. Logarithmic differentiation produces:
For a general u(x)v(x), the calculator uses uv[v′ ln(u) + v·u′/u]. This formula uses the real natural logarithm and therefore assumes u > 0 on the interval being considered. A constant negative base raised to a variable real exponent is not one continuous real-valued function, so the calculator rejects that symbolic case.
Constant rational powers receive the ordinary power rule. For example, the derivative of x1/3 is (1/3)x-2/3. The point evaluator preserves an explicitly entered rational exponent, which permits real odd roots at negative x-values while still rejecting even roots of negative values.
First, Second and Third Derivatives
The second derivative differentiates the first derivative. The third derivative differentiates the second. Different notation highlights the order:
- First derivative: f′(x) or dy/dx
- Second derivative: f″(x) or d2y/dx2
- Third derivative: f‴(x) or d3y/dx3
For position s(t), the first derivative is velocity and the second derivative is acceleration. For a graph, the sign of the second derivative helps describe concavity where the function is twice differentiable. A positive second derivative indicates local concave-up behavior, while a negative value indicates local concave-down behavior.
A higher derivative may become too large to present clearly when the starting expression contains many nested products, quotients or powers. The calculator caps expression size and differentiation complexity to keep the Blogger page responsive.
Derivative at a Point, Tangent Line and Normal Line
When you enter x = a, the calculator substitutes a into both the original function and the selected derivative. For a first derivative, it also builds tangent and normal lines if the required values are finite.
If the tangent slope is zero, the normal line is vertical and has the equation x = a. If the function or derivative is undefined at a, no valid tangent calculation is shown. The symbolic derivative may still be useful on other points in the function’s domain.
For a second or third derivative selection, the numerical panel evaluates that selected derivative. Tangent and normal lines belong to the first derivative, so the line fields direct you to select First derivative.
Domain Limits and Nondifferentiable Points
A symbolic formula does not guarantee that every real x-value is valid. Check the domain of the original function and the derivative. Common restrictions include:
- A denominator cannot equal zero.
- ln(x) and log(x) require a positive real argument.
- sqrt(x) requires a nonnegative real argument, and its standard derivative formula has a zero denominator at x = 0.
- asin(x) and acos(x) accept values from -1 to 1, but their derivatives are not finite at the endpoints.
- abs(x) is continuous at zero but has different left and right slopes, so it is not differentiable there.
- A corner, cusp, jump or vertical tangent may prevent a finite two-sided derivative.
Supported Input and Calculator Limits
Use x as the variable. The expression engine supports +, -, *, /, ^, parentheses, pi, e and the named functions listed on this page. It accepts common implicit multiplication, but it does not accept equations with both x and y, integrals, limits, piecewise syntax, factorials, floor or ceiling derivatives, complex numbers or user-defined functions.
This scope is deliberate. Discontinuous and piecewise functions need interval conditions. An equation such as x2 + y2 = 25 needs implicit differentiation. A parametric or multivariable function needs a different derivative definition. Use an appropriate specialist tool when the mathematical structure falls outside an explicit function f(x).
Inputs longer than 180 characters, unsupported names, malformed scientific notation, unmatched parentheses, excessive nesting and oversized derivative trees receive controlled errors. The engine never runs the expression as JavaScript.
Common Differentiation Mistakes
- Reducing the exponent but forgetting to multiply by the original exponent in the power rule.
- Differentiating a product as u′v′ instead of u′v + uv′.
- Differentiating a quotient as u′/v′ instead of applying the quotient rule.
- Forgetting the derivative of the inner function in the chain rule.
- Using degree-mode trigonometric values with derivative formulas derived for radians.
- Substituting a point that lies outside the original function’s domain.
- Assuming a simplified formula restores a point excluded from the original expression.
- Rounding an intermediate slope before building the tangent or normal line.
Use the displayed rules to check the structure of your work. For graded or professional calculations, write the required reasoning in the notation expected by your course or field.
Where Derivatives Are Used
- Motion: Derivatives convert position to velocity and velocity to acceleration.
- Optimization: Critical points help locate candidate maximum and minimum values.
- Economics: Marginal cost and marginal revenue measure approximate change per additional unit.
- Engineering: Sensitivity and response rates describe how outputs react to changing inputs.
- Science: Rates model population change, reactions, heat transfer and other dynamic processes.
- Data methods: Gradients guide many numerical fitting and machine-learning procedures.
A correct derivative does not prove that a model, unit choice or input range is appropriate. Verify the assumptions and interpret the sign and units in the context of the original problem.
Accuracy, Privacy and Review Basis
The symbolic engine uses deterministic differentiation rules and bounded simplification. Point evaluation uses standard browser number arithmetic, which provides about 15 to 17 significant decimal digits for many finite values. Transcendental functions and some fractions need rounded decimal displays. The precision setting affects displayed values only.
The derivative definition, tangent interpretation and elementary rules follow OpenStax Calculus Volume 1: Defining the Derivative and Differentiation Rules. Nested-function handling follows the OpenStax Chain Rule.
Your function and point stay in the current browser page. Reloading clears them. Report a repeatable issue through the 1Dollars contact page with the exact expression, derivative order, x-value and precision used.
Derivative Calculator vs Related Math Tools
| Tool | Best use | Main output |
|---|---|---|
| Derivative Calculator | Differentiate an explicit function of x | Symbolic derivative, point value and steps |
| Graphing Calculator | Inspect function shape, intercepts and local behavior | Plot, trace and value table |
| Algebra Calculator | Simplify or evaluate supported algebra expressions | Algebraic result and steps |
| Equation Calculator | Solve supported one-variable equations | Exact and decimal solutions |
| Scientific Calculator | Evaluate numerical trig, log, root and power expressions | Numerical result |
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Frequently Asked Questions
What does a derivative calculator find?
It converts a supported function f(x) into its first, second or third derivative. This tool also evaluates the result at a point and shows tangent information for a first derivative.
Does the calculator show differentiation steps?
Yes. It identifies the applicable constant, power, sum, product, quotient, chain and elementary-function rules, then shows the simplified derivative and point substitution.
How do I enter x squared or a square root?
Enter x^2 for x squared. Enter sqrt(x) for a square root, or use a rational power such as x^(1/2).
Does the calculator use degrees or radians?
It uses radians. Standard trigonometric derivative formulas assume radian measure, so no degree-mode option is provided.
Can it calculate second and third derivatives?
Yes. Choose the required order. The calculator differentiates the preceding result again and reports the selected higher derivative.
Why is the derivative undefined at my x-value?
The point may lie outside the function’s domain or at a corner, cusp, zero denominator or other nondifferentiable location. The symbolic formula may remain valid elsewhere.
Can it differentiate x raised to x?
Yes. It uses logarithmic differentiation and returns x^x times [ln(x) + 1] on the standard positive real domain.
Does it support implicit differentiation?
No. Enter an explicit function of x. Relations containing both x and y, parametric curves and multivariable functions need a specialist calculator.
How is the tangent line calculated?
For x = a, the tool uses the point (a, f(a)) and slope f′(a) in y - f(a) = f′(a)(x - a).
Are entered functions stored or sent to a server?
No calculation server receives the expression. Parsing, symbolic differentiation and point evaluation run in the current browser page.
Educational and financial disclaimer: This calculator provides general mathematical results from your inputs. It is not financial, scientific, engineering, tax or professional advice. Verify functions, domains, units, assumptions and required precision before using a result for an important decision.