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Derivative calculator

Find a derivative

Differentiate one function with respect to a chosen variable and see which rule applies at each step.

Use the math keyboard or your device keyboard.
Try an example

Result

First derivative

2xsin⁡(x)+x2cos⁡(x)2x\sin(x)+x^2\cos(x)
Conditions
  • x is real.
  • The result is defined for every real x.

Steps

  1. Select the product rule. Two differentiable factors are multiplied.(fg)′=f′g+fg′(fg)'=f'g+fg'
  2. Differentiate each factor. Use the power rule and the sine derivative.(x2)′=2x,(sin⁡x)′=cos⁡x(x^2)'=2x,\quad (\sin x)'=\cos x
  3. Combine without dropping a factor. Each term keeps the factor that was not differentiated.2xsin⁡x+x2cos⁡x2x\sin x+x^2\cos x
Independent check

A central-difference estimate of the original function agrees with the symbolic derivative at regular sample points.

Scope

What this derivative calculator covers

The derivative calculator handles explicit single-variable expressions. It preserves derivative order and variable choices instead of guessing from ambiguous input.

Basic derivative rules

Constants, powers, sums, and scalar multiples form the base of longer derivatives.

Examples: polynomials, negative powers, fractional powers on their real domains

Products and quotients

Preserve both original factors while applying the product or quotient rule.

Examples: x squared times sine x, exponential over polynomial

Compositions

Apply the chain rule from the outside function toward the inside function.

Examples: sine of x squared, logarithm of a polynomial

Higher derivatives

Repeat differentiation for the requested order and keep the order visible in the interpreted task.

Examples: second derivative, third derivative

How to use it

Enter enough information for one clear task

  1. 1
    Enter one explicit function

    Use parentheses around function arguments and denominators so the structure is unambiguous.

  2. 2
    Choose the variable and order

    A symbol that is not the differentiation variable is treated as constant only when the input makes that intent clear.

  3. 3
    Check the rule sequence

    Read why the product, quotient, or chain rule was chosen before following the algebra.

  4. 4
    Compare domains

    A derivative formula can be algebraically correct but valid on a smaller domain than a casual simplification suggests.

Worked inputs

Examples to try

Use these examples to recognize the method, compare equivalent forms, and check your own work.

Power rule

linearity

ddx(3x5−2x+7)\frac{d}{dx}(3x^5-2x+7)

Expected result

15x4−215x^4-2

Product rule

product rule

ddx(x2ex)\frac{d}{dx}(x^2e^x)

Expected result

ex(x2+2x)e^x(x^2+2x)

Quotient rule

quotient rule

ddx(xx+1)\frac{d}{dx}\left(\frac{x}{x+1}\right)

Expected result

1(x+1)2\frac{1}{(x+1)^2}

Chain rule

chain rule

ddxsin⁡(x2)\frac{d}{dx}\sin(x^2)

Expected result

2xcos⁡(x2)2x\cos(x^2)

Logarithm

chain rule

ddxln⁡(x2+1)\frac{d}{dx}\ln(x^2+1)

Expected result

2xx2+1\frac{2x}{x^2+1}

Second derivative

repeat the power rule

d2dx2x4\frac{d^2}{dx^2}x^4

Expected result

12x212x^2

Complete example

Differentiate an implicit circle

Treat y as a function of x. Every derivative of a y-expression therefore includes dy over dx.

x2+y2=25x^2+y^2=25
  1. 1
    Differentiate both sides

    The constant differentiates to zero.

    2x+2ydydx=02x+2y\frac{dy}{dx}=0
  2. 2
    Isolate the y derivative

    Move 2x to the other side and divide by 2y.

    dydx=−xy\frac{dy}{dx}=-\frac{x}{y}
  3. 3
    State the local condition

    This slope expression requires y not equal to zero; the circle has vertical tangents at those points.

    y≠0y\ne0
dydx=−xy\frac{dy}{dx}=-\frac{x}{y}

Verification: At the point (3,4), a local parametrization of the circle gives slope minus 3 over 4, matching the implicit result.

Avoidable errors

Common mistakes and how to fix them

Dropped inner derivative

Problem: Writing the derivative of sine of x squared as cosine of x squared.

Why it matters: The outside derivative must be multiplied by the derivative of x squared.

Better approach: Write the outer derivative first, then multiply by 2x.

Incomplete product rule

Problem: Differentiating both factors and multiplying the derivatives.

Why it matters: The product rule is not f prime times g prime.

Better approach: Use f prime g plus f g prime.

Sign error in quotient rule

Problem: Reversing the numerator subtraction.

Why it matters: Changing f prime g minus f g prime changes the result.

Better approach: Name the numerator f and denominator g before substituting.

Ignoring the domain

Problem: Cancelling factors without recording excluded values.

Why it matters: An equivalent simplified formula may not restore points missing from the original function.

Better approach: Carry original domain restrictions into the answer.

Trust the result for the right reasons

Checks, assumptions, and limits

How results are checked

  • Simplify the result independently and compare equivalent forms.
  • Substitute regular sample values when a numerical check is meaningful.
  • Check the domain and any one-sided, endpoint, or constant assumptions.

When to stop and revise the input

  • Revise an expression when the variable, boundary, direction, or grouping is ambiguous.
  • Treat a symbolic result as unsupported when the engine cannot establish an elementary form.
  • Use the steps to check work, not as a substitute for course or examination rules.

Common questions

Find a derivative FAQ

Does the calculator show steps?

Yes. A result includes the interpreted problem, the final answer, the mathematical operation used, relevant conditions, and a verification note when an independent check is available.

Why can a correct answer look different from a textbook answer?

Equivalent algebraic and trigonometric forms can look different while representing the same function. Indefinite integrals can also differ by a constant. Compare by simplifying, differentiating, or substituting values within the shared domain.

What should I include in the input?

State the expression, variable, and operation. Add an interval, upper and lower bounds, a one-sided direction, derivative order, or initial condition whenever the task depends on it.

How should I check an important result?

Inspect the recognized problem and assumptions first. Then use an independent check such as differentiating an antiderivative, substituting a solution into the original equation, or comparing one-sided numerical values.

Continue learning

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Created by Mathos AI. Methods, conditions, and checks are shown so you can review the mathematical reasoning.