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Derivative Calculator

Differentiate nine polynomial, power, trigonometric, exponential, logarithmic, radical, and reciprocal function families with evaluation steps.

  • differentiation calculator
  • derivative rules calculator
  • trigonometric derivative calculator
  • exponential derivative calculator
  • derivative at a point

Use terms such as 3x^2 + 2x - 5. Decimal and negative exponents are supported.

The point at which the derivative value is calculated.

Status: initial

Results

Awaiting calculation

Explore your result

Derivative as slope

Compare the calculated derivative value while keeping the selected function, symbolic rule, and substitution steps visible.

Derivative as slope
Slope at x12

Use this result well

  1. 1Choose the matching function family.
  2. 2Check coefficients, exponent, and evaluation point.
  3. 3Review the symbolic rule and substituted value together.

The derivative is the instantaneous slope at one point; trigonometric derivative rules use radians.

Calculus guide

Derivative calculator for nine function families

Choose a supported function family, enter its parameters, and calculate both the symbolic first derivative and its value at a selected x. The form changes to show only inputs used by that rule. The guided function selector keeps polynomial, trigonometric, exponential, logarithmic, radical, and reciprocal rules explicit while evaluating the derivative at a selected x-value.

When to use this derivative calculator

Use it to check introductory calculus work involving polynomials, powers, trigonometric functions, e-based exponentials, natural logarithms, square roots, and reciprocals. It is a guided rule calculator rather than an unrestricted computer algebra system. Use it to verify introductory calculus exercises and tangent slopes across the supported families. It does not claim unrestricted product-rule, quotient-rule, nested-chain, or arbitrary symbolic algebra support.

How differentiation is applied

Polynomial terms use the power rule. Trigonometric and exponential families apply the chain rule to bx. Logarithm, square-root, and reciprocal modes enforce the real-number domain required at the selected x value. Power functions reduce the exponent by one, while trigonometric and exponential functions apply the chain-rule multiplier from bx; each result identifies the rule used.

Variable explanations

Understand what each input and result means before calculating.

Function family

Selects the derivative rule and the relevant parameter fields.

Coefficient a

An outer multiplier carried through differentiation.

Exponent n

Power in a·xⁿ; it may be positive, negative, or fractional where the real domain permits.

Inner coefficient b

Multiplier inside bx that appears through the chain rule.

Evaluate at x

The input point used to calculate the numerical derivative value.

Reviewed by the Calculator.org.in Editorial Team

Formula behavior, validation cases, explanatory examples, and cited sources are checked before publication. This review supports educational accuracy and is not a substitute for qualified professional advice.

Last reviewed: 2026-08-23

Review process

Formula guide

See the calculation logic, variable definitions, and practical meaning.

Polynomial and power rules

d(axⁿ)/dx = anxⁿ⁻¹

  • a is coefficient.
  • n is exponent.

Apply the rule to every polynomial term; constants become zero.

Sine

d[a sin(bx)]/dx = ab cos(bx)

  • Angles are in radians.

The inner derivative b multiplies the outer derivative.

Cosine

d[a cos(bx)]/dx = −ab sin(bx)

  • Angles are in radians.

Cosine introduces a negative sign and the chain-rule factor b.

Tangent

d[a tan(bx)]/dx = ab sec²(bx)

  • cos(bx) cannot equal zero.

Tangent and its derivative are undefined where cosine is zero.

Exponential

d[ae^(bx)]/dx = ab e^(bx)

  • e is Euler's number.

An e-based exponential retains its form and gains the inner factor b.

Natural logarithm

d[a ln(x)]/dx = a/x

  • x must be greater than zero for this real-valued mode.

The natural logarithm's slope decreases as x increases.

Square root and reciprocal

d(a√x)/dx = a/(2√x); d(a/x)/dx = −a/x²

  • Square-root derivative requires x > 0.
  • Reciprocal requires x ≠ 0.

Both follow from rewriting the function with fractional or negative powers.

Worked examples

Follow realistic inputs through the calculation step by step.

1

Worked example

Polynomial

  1. 1Choose Polynomial and enter 3x^2 + 2x - 5.
  2. 2Apply the power rule term by term.
  3. 3The derivative is 6x + 2; at x = 2 it equals 14.
2

Worked example

Sine with chain rule

  1. 1Choose Sine with a = 3 and b = 2.
  2. 2The derivative of 3sin(2x) is 6cos(2x).
  3. 3Substitute the chosen x using radians.
3

Worked example

Exponential

  1. 1Choose Exponential with a = 4 and b = 0.5.
  2. 2Differentiate 4e^(0.5x).
  3. 3The derivative is 2e^(0.5x).
4

Worked example

Natural logarithm

  1. 1Choose Natural logarithm with a = 5.
  2. 2Use d[5ln(x)]/dx = 5/x.
  3. 3At x = 10 the derivative is 0.5.
5

Worked example

Differentiate a cubic polynomial

  1. 1Enter 2x^3 - 4x^2 + 7.
  2. 2Differentiate each term with the power rule.
  3. 3The derivative is 6x² - 8x.
6

Worked example

Evaluate a tangent slope

  1. 1Enter x^2 - 3x + 2 and set x to 4.
  2. 2The derivative is 2x - 3.
  3. 3At x = 4, the derivative and tangent slope equal 5.

Common mistakes

Avoid these common input and interpretation errors.

Using degrees for derivatives

Standard trigonometric derivative rules assume radian measure.

Forgetting the chain rule

Functions containing bx gain a factor of b.

Dropping negative signs

The derivative of cosine and reciprocal functions includes a negative sign.

Ignoring domains

Logarithm, square root, reciprocal, tangent, and fractional powers can be undefined at some real x values.

Expecting arbitrary syntax

Choose a supported family; only Polynomial mode accepts a typed expression.

Frequently asked questions

Quick answers to the questions users ask most often.

Which derivative functions are supported?
Polynomial, power, sine, cosine, tangent, e-based exponential, natural logarithm, square root, and reciprocal functions.
Does it apply the chain rule?
Yes. Sine, cosine, tangent, and exponential modes include the inner coefficient b from bx.
Are trigonometric inputs degrees or radians?
Radians. The standard calculus identities for sine, cosine, and tangent derivatives assume radians.
Can I enter any expression?
No. Polynomial mode accepts polynomial syntax; other modes use structured parameters for transparent, reliable calculations.
Why can a derivative be undefined at x?
Some functions have domain restrictions or vertical behavior. The calculator reports these cases instead of returning a misleading finite value.
What does derivative at x mean?
It is the instantaneous rate of change and tangent-line slope at the selected input when the derivative exists.
What does a derivative represent?
At a point, a derivative represents instantaneous rate of change and the slope of the tangent line when that derivative exists.
Why does the exponent decrease by one?
That is the polynomial power rule: differentiating xⁿ gives n·xⁿ⁻¹. The original exponent also becomes a coefficient multiplier.
Can the derivative be zero at a point?
Yes. A zero derivative indicates a horizontal tangent, though additional analysis is needed to decide whether the point is a maximum, minimum, or neither.

References

Sources used to support the calculator guidance.

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Version history

A transparent record of calculator content updates.

Updated 2026-08-23
  • 2.0.0 · 2026-08-23

    Expanded from polynomial-only differentiation to nine guided function families with conditional inputs, chain-rule support, domains, symbolic results, numerical evaluation, and steps.

  • 1.0.0 · 2026-08-03

    Initial polynomial power-rule calculator.