Derivative Calculator

Differentiate at a point

Build the derivative

Enter one real function of x, choose the derivative order, and set the point. The preview keeps the complete question visible.

1st–3rd orderOne real variableTrig uses radians

Derivative previewFirst derivative with respect to x

x³at x = 2

Function f(x)Type the function only. Use * for multiplication, ^ for powers, and parentheses for functions such as sin(x) or exp(x).

Evaluate at xThe graph centers on this point.

Click to insert

Formula templates

Insert at the cursor, or select part of the function first to wrap it. New parameters are highlighted for quick replacement.

Structure

Functions & constants

Derivative order

Higher orders repeat differentiation. A tangent line is shown only for the first derivative.

Example derivativesLoad a complete function, point, and order.

Press Enter from the function or point to calculate.

Derivative result

Answer and graph

Ready

Production-authorized result · formula 1.0.0

Bounded symbolic derivativeFrom f(x) = x³

f′(x)3 · (x²)

The symbolic expression is shown first; the point value beside it evaluates this derivative at x = 2.

First derivative at x = 2

f′(2)12.00

This is the instantaneous slope of f(x) at x = 2.

Coordinate graph

Function and derivative near x = 2

The plots use separate y-scales so the function and derivative remain readable. The vertical guide marks the evaluation point.

Original function f(x)f(2) = 8

First derivative f′(x)f′(2) = 12

f(x) f′(x) Tangent at x = 2

Read the result

Function value, tangent, and method

Function value f(2)8.00

Tangent liney = 8 + 12 * (x - 2)

MethodSymbolic rule subset with finite-difference cross-check

Show calculation steps

Tokenize and parse the real function with the shared bounded expression grammar.f(x) = x^3; trig unit = radians
Apply the supported rules repeatedly or use a fixed finite-difference fallback when the symbolic subset declines the expression.order = 1; derivative = (3 * (x ^ 2))
Evaluate at the requested point and compare bounded symmetric stencil refinements.f^(1)(2) = 12

Numerical cross-check

Difference / refinement indicator0.00

The table checks the result at several step sizes. It supports numerical stability but does not by itself prove differentiability.

Drawing the cross-check table…

Scope and important assumptions

The variable and derivative are unit-neutral. If x and f(x) carry units, the derivative unit is output-unit per input-unit to the selected order; no conversion is inferred.
Trigonometric and inverse-trigonometric values use radians. The displayed derivative is intentionally unsophisticated and may be algebraically correct without being fully simplified.
A finite-difference table is evidence of numerical stability at sampled scales, not a proof of differentiability. Piecewise definitions, sharp cusps, domain boundaries, discontinuities, severe cancellation, and functions outside the symbolic subset can require manual analysis.
This tool does not provide the full symbolic scope or step engine of Symbolab or Wolfram|Alpha.

Local rate and curvature

Differentiate a real function at one declared point

A derivative answer belongs to a function, an evaluation point, and an order. The local workbench pairs bounded symbolic rules with numerical cross-checks so those three choices remain auditable.

Step-by-step guide · Local rate and curvature

Work a cubic at x equals two

Available orders: First, second, or third
Primary interpretation: Local rate of change
Trigonometric convention: Radians

Compile the explicit function
The bounded parser accepts the documented real expression language and records a normalized formula. It does not execute arbitrary code or interpret implicit equations.
Apply the requested rule order
The supported symbolic subset applies constant, sum, product, quotient, chain, power, exponential, logarithmic, and trigonometric rules. For x^3, one application yields 3x^2.
Evaluate function and derivative
At x=2, f(2)=8 and f′(2)=12. The first-order result also produces tangent form y=8+12(x−2).
Read the finite-difference cross-check
Symmetric stencil refinements compare local numerical slopes at several scales. Agreement is useful evidence of numerical stability, but it is not a proof of differentiability.

Differentiate a real function at one declared point: For f(x)=x^3 and a=2, the first derivative should be 3x^2 and the local slope should be 12.

Name the local question before differentiating

For position s(t), the first derivative is velocity and the second is acceleration when units and smoothness support those interpretations. For a response curve f(x), f′(a) is the limiting slope of secant lines near a. The number says nothing about distant inputs, and a third derivative answers a different question from a first derivative even when both are evaluated at the same point.

Compare two nearby sampled points with the finding slope calculator to make the secant-rate idea concrete before taking a local limit. A sharp corner, jump, endpoint, or vertical behavior is a warning that a finite tangent slope may not exist, no matter how plausible a nearby numerical estimate looks.

Work a cubic at x equals two

For f(x)=x^3 and a=2, the first derivative should be 3x^2 and the local slope should be 12.

Compile the explicit function
The bounded parser accepts the documented real expression language and records a normalized formula. It does not execute arbitrary code or interpret implicit equations.
Apply the requested rule order
The supported symbolic subset applies constant, sum, product, quotient, chain, power, exponential, logarithmic, and trigonometric rules. For x^3, one application yields 3x^2.
Evaluate function and derivative
At x=2, f(2)=8 and f′(2)=12. The first-order result also produces tangent form y=8+12(x−2).
Read the finite-difference cross-check
Symmetric stencil refinements compare local numerical slopes at several scales. Agreement is useful evidence of numerical stability, but it is not a proof of differentiability.

Keep derivative, limit, and integral roles separate

Derivative at a point

This page evaluates a local rate and, for first order, a tangent-line form. It does not search globally for extrema or solve differential equations.

Limit toward a point

Differentiability is defined through a limit. Use the limit calculator to inspect a separate approach problem, remembering that numerical convergence is still not an epsilon-delta proof.

Accumulation across an interval

The definite integral calculator accumulates signed contribution from a to b. A derivative value at one point cannot substitute for that interval calculation.

Stop where local numerical evidence becomes unreliable

Absolute value may be handled numerically only in a constrained continuous fallback, while floor, ceiling, rounding, and sign expressions are rejected to avoid misleading derivatives at jumps.
The displayed symbolic derivative can be correct without being fully simplified. Algebraic appearance is not a ranking of mathematical quality.
Partial, implicit, parametric, distributional, complex, and arbitrary-order derivatives are outside scope. The supported order stops at three for one explicit real variable.
Units remain the user's responsibility. A first derivative has output-unit per input-unit; higher orders change that dimensional relationship and should not be reported as unitless by habit.

Important limitations

Derivative Calculator is limited to the finite inputs, modes, and outputs stated for the Derivative at a point method. Check function f(x), evaluate at x, and derivative order, validation messages, the equation, and the stated display precision before relying on derivative at the point, function value f(a), bounded symbolic derivative, and method used.

This formula passed its registered engineering tests and is published under an explicit site-owner policy authorization. No independent expert, QA, editorial, legal, or third-party review is claimed. Verify consequential results with a qualified professional.

Step h

Derivative estimate

Change from prior

0.0118415356759

—

0.00592076783793

9.41469125e-13

0.00296038391897

5.20472554e-13

0.00148019195948

3.10507176e-12

0.000740095979741

6.73239242e-12

0.000370047989871

1.8115287e-11

Local rate and curvature

Differentiate a real function at one declared point

A derivative answer belongs to a function, an evaluation point, and an order. The local workbench pairs bounded symbolic rules with numerical cross-checks so those three choices remain auditable.

Step-by-step guide · Local rate and curvature

Work a cubic at x equals two

Available orders: First, second, or third
Primary interpretation: Local rate of change
Trigonometric convention: Radians

Compile the explicit function
The bounded parser accepts the documented real expression language and records a normalized formula. It does not execute arbitrary code or interpret implicit equations.
Apply the requested rule order
The supported symbolic subset applies constant, sum, product, quotient, chain, power, exponential, logarithmic, and trigonometric rules. For x^3, one application yields 3x^2.
Evaluate function and derivative
At x=2, f(2)=8 and f′(2)=12. The first-order result also produces tangent form y=8+12(x−2).
Read the finite-difference cross-check
Symmetric stencil refinements compare local numerical slopes at several scales. Agreement is useful evidence of numerical stability, but it is not a proof of differentiability.

Differentiate a real function at one declared point: For f(x)=x^3 and a=2, the first derivative should be 3x^2 and the local slope should be 12.

Name the local question before differentiating

Work a cubic at x equals two

For f(x)=x^3 and a=2, the first derivative should be 3x^2 and the local slope should be 12.

Compile the explicit function
The bounded parser accepts the documented real expression language and records a normalized formula. It does not execute arbitrary code or interpret implicit equations.
Apply the requested rule order
The supported symbolic subset applies constant, sum, product, quotient, chain, power, exponential, logarithmic, and trigonometric rules. For x^3, one application yields 3x^2.
Evaluate function and derivative
At x=2, f(2)=8 and f′(2)=12. The first-order result also produces tangent form y=8+12(x−2).
Read the finite-difference cross-check
Symmetric stencil refinements compare local numerical slopes at several scales. Agreement is useful evidence of numerical stability, but it is not a proof of differentiability.

Keep derivative, limit, and integral roles separate

Derivative at a point

This page evaluates a local rate and, for first order, a tangent-line form. It does not search globally for extrema or solve differential equations.

Limit toward a point

Differentiability is defined through a limit. Use the limit calculator to inspect a separate approach problem, remembering that numerical convergence is still not an epsilon-delta proof.

Accumulation across an interval

The definite integral calculator accumulates signed contribution from a to b. A derivative value at one point cannot substitute for that interval calculation.

Stop where local numerical evidence becomes unreliable

Absolute value may be handled numerically only in a constrained continuous fallback, while floor, ceiling, rounding, and sign expressions are rejected to avoid misleading derivatives at jumps.
The displayed symbolic derivative can be correct without being fully simplified. Algebraic appearance is not a ranking of mathematical quality.
Partial, implicit, parametric, distributional, complex, and arbitrary-order derivatives are outside scope. The supported order stops at three for one explicit real variable.
Units remain the user's responsibility. A first derivative has output-unit per input-unit; higher orders change that dimensional relationship and should not be reported as unitless by habit.

Derivative Calculator

Build the derivative

Formula templates

Answer and graph

Function and derivative near x = 2

Function value, tangent, and method

Differentiate a real function at one declared point

Work a cubic at x equals two

Compile the explicit function

Apply the requested rule order

Evaluate function and derivative

Read the finite-difference cross-check

Name the local question before differentiating

Work a cubic at x equals two

Keep derivative, limit, and integral roles separate

Derivative at a point

Limit toward a point

Accumulation across an interval

Stop where local numerical evidence becomes unreliable

Important limitations

Build the derivative

Formula templates

Answer and graph

Function and derivative near x = 2

Function value, tangent, and method

Finite-difference cross-check

Differentiate a real function at one declared point

Work a cubic at x equals two

Compile the explicit function

Apply the requested rule order

Evaluate function and derivative

Read the finite-difference cross-check

Name the local question before differentiating

Work a cubic at x equals two

Keep derivative, limit, and integral roles separate

Derivative at a point

Limit toward a point

Accumulation across an interval

Stop where local numerical evidence becomes unreliable

Finite-difference cross-check