Derivative Calculator

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Step-by-step hints

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Worked example

Step-by-step solution

$\frac{d}{dx}\left[x^2 \sin x\right]$

Step 1 / 3

Apply the product rule

The function is a product of f = x² and g = sin x.

\frac{d}{dx}[fg] = f'g + fg'

How to use the Derivative Calculator

Enter a function and this derivative calculator differentiates it step by step, naming the power, product, quotient or chain rule each time it is used and simplifying the result.

Type the function, for example x² sin(x). Use ^ for powers and parentheses for function arguments.
Press Solve it to differentiate with respect to x (say “with respect to t” to change the variable).
Each step title names the rule applied, so you can match it to your class notes.
The final step simplifies the derivative.

The rules behind every derivative

(fg)' = f'g + fg'

Most derivatives come from a few rules. The power rule handles xⁿ, the product rule handles f·g, the quotient rule handles f/g and the chain rule handles a function inside another function.

Recognizing the outermost structure tells you which rule to apply first. For x² sin(x) that structure is a product, so the product rule comes first and the power rule and the derivative of sine fill in the pieces.

Derivative calculator example: the chain rule

\frac{d}{dx}\ln(3x^2+1)=\frac{6x}{3x^2+1}

For ln(3x² + 1), identify two layers before taking the derivative. The outer function is ln(u), whose derivative is 1/u. The inner function is u = 3x² + 1, whose derivative is 6x. Multiply the outer derivative, evaluated at the inner expression, by the inner derivative to obtain 6x/(3x² + 1). Forgetting that final multiplication is a common chain-rule error.

The derivative calculator should show where the factor 6x comes from, rather than only presenting a fraction. The logarithm requires a positive argument; here 3x² + 1 is positive for every real x, so both the original function and its derivative are defined for all real inputs. For a different logarithm, check its argument before substituting a value.

Derivative calculator example: the quotient rule

\frac{d}{dx}\frac{x+1}{x-1}=\frac{(x-1)-(x+1)}{(x-1)^2}=\frac{-2}{(x-1)^2}

For (x + 1)/(x − 1), use the quotient rule: multiply the derivative of the numerator by the original denominator, subtract the original numerator times the derivative of the denominator, and divide by the denominator squared. Both simple derivatives equal 1, leaving (x − 1) − (x + 1) in the numerator. Distribute the subtraction sign carefully to obtain −2.

The original function is undefined at x = 1, and the derivative is also undefined there. Keep that restriction with the simplified answer. Differentiating the numerator and denominator independently and dividing their derivatives would give 1, which is incorrect. The quotient-rule steps explain why the changing denominator contributes to the result.

Evaluate a derivative at a point or differentiate again

\frac{d^2}{dx^2}(x^2\sin x)=2\sin x+4x\cos x-x^2\sin x

To find the slope of x² sin(x) at x = π, first differentiate the function and then substitute π into 2x sin(x) + x² cos(x). Since sin(π) = 0 and cos(π) = −1, the slope is −π². Substituting into the original function would give its height, not its slope. For the second derivative, apply the product rule to each term of the first derivative and combine the two 2x cos(x) terms into 4x cos(x).

Frequently asked questions

Can it find second derivatives?

Yes. Ask for “the second derivative of …” and it differentiates twice, showing both rounds.

Does it handle implicit differentiation?

Yes. Enter an equation such as x² + y² = 25 and ask for dy/dx; the steps differentiate both sides and solve for dy/dx.

Can I evaluate the derivative at a point?

Add the point, for example “derivative of x² sin(x) at x = π”, and the last step substitutes it.

Have a different kind of math problem? Try the free math solver. Use the subject menu for physics and chemistry.