Calculator

Derivative Calculator with Steps

Differentiate polynomials, roots, exponentials, logs, and trig functions, and see the rule that produced every line — including the inner derivative that the chain rule keeps demanding.

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How to use it

Type the function the way you would write it by hand: x^3*ln(x), sin(2x), and e^(2x)/(x^2+1) all parse. If you are differentiating with respect to something other than $x$ , name the variable in the question. State whether you want a first, second, or implicit derivative, because those are three different pieces of work. Every step is labelled with its rule, and follow-up questions on the same solution do not use up your daily quota, so ask why a line happened rather than resolving the whole problem. For mixed problems that also involve limits or integrals, the calculus solver handles the whole question in one pass.

The method behind it

A derivative is the instantaneous rate of change of a function: the limit of average rates of change as the measuring interval shrinks to nothing.

f^{'} (x) = h \to 0 lim \frac{f ( x + h ) - f ( x )}{h}

Every rule below is that limit already worked out for a whole family of functions, which is why nobody differentiates from the definition after the first week of the course. The power rule handles any real exponent, so rewrite roots and reciprocals before you start — $x = x^{1/2}$ and $1/ x^{3} = x^{- 3}$ .

\frac{d}{d x} x^{n} = n x^{n - 1}

Two functions multiplied need the product rule, and the structure is symmetric: differentiate one factor at a time and add the results.

\frac{d}{d x} [f (x) g (x)] = f^{'} (x) g (x) + f (x) g^{'} (x)

The quotient rule is the same idea with a subtraction, and the subtraction is the reason it goes wrong so often. The derivative of the numerator comes first.

\frac{d}{d x} [\frac{f ( x )}{g ( x )}] = \frac{f ^{'} ( x ) g ( x ) - f ( x ) g ^{'} ( x )}{[ g ( x ) ] ^{2}}

The chain rule is the one that separates people who can differentiate from people who can differentiate reliably. Any time the variable is wrapped inside another function, the derivative of that inner expression must multiply what you wrote.

\frac{d}{d x} f (g (x)) = f^{'} (g (x)) \cdot g^{'} (x)

Those four rules compose. A quotient whose numerator is a composite needs the quotient rule on the outside and the chain rule inside the top, and the order you apply them in is fixed by the structure of the expression, not by preference. Layer them onto the standard library of derivatives below and almost everything in a first course is reachable.

\frac{d}{d x} sin x = cos x, \frac{d}{d x} cos x = - sin x, \frac{d}{d x} e^{x} = e^{x}, \frac{d}{d x} ln x = \frac{1}{x}

If the composition is what is tripping you up, work through the chain rule explained plainly before returning here; the product and quotient rules guide covers the sign error that costs the most marks.

Worked examples

Worked example

\frac{d}{d x} (x^{3} ln x)

1
The outermost operation is a multiplication of two functions of x, so this is the product rule.
$f (x) = x^{3}, g (x) = ln x$
2
Differentiate each factor separately before combining anything.
$f^{'} (x) = 3 x^{2}, g^{'} (x) = \frac{1}{x}$
3
Apply the rule: derivative of the first times the second, plus the first times the derivative of the second.
$3 x^{2} ln x + x^{3} \cdot \frac{1}{x}$
4
Simplify the second term. Three powers of x over one power of x leaves two.
$3 x^{2} ln x + x^{2}$
5
Factor out the common square so the answer is in its tidiest form.
$x^{2} (3 ln x + 1)$

Answer

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

Worked example

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

1
The last operation is a division and the denominator has two terms, so the quotient rule applies. Name the top and bottom first.
$f (x) = e^{2 x}, g (x) = x^{2} + 1$
2
The numerator is a composite, so its derivative needs the chain rule: differentiate the exponential, then multiply by the derivative of the exponent.
$f^{'} (x) = 2 e^{2 x}, g^{'} (x) = 2 x$
3
Substitute into the quotient rule, keeping the derivative of the top in the first product.
$\frac{2 e ^{2 x} ( x ^{2} + 1 ) - e ^{2 x} ( 2 x )}{( x ^{2} + 1 ) ^{2}}$
4
Both terms on top share a factor of 2e^{2x}. Pull it out rather than expanding.
$\frac{2 e ^{2 x} ( x ^{2} + 1 - x )}{( x ^{2} + 1 ) ^{2}}$
5
Order the bracket by descending powers. The quadratic inside has a negative discriminant, so it never factors further and never vanishes.
$\frac{2 e ^{2 x} ( x ^{2} - x + 1 )}{( x ^{2} + 1 ) ^{2}}$

Answer

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

Which rule applies?

Choosing the rule is the whole difficulty; executing it is bookkeeping. The question that resolves it every time is which operation you would perform last if you were plugging a number in. Answer that, and the rule is forced.

Pick the differentiation rule

Imagine evaluating the expression at a number. Which operation would you perform last?

Common mistakes

Dropping the inner derivative

Writing $\frac{d}{d x} sin (3 x) = cos (3 x)$ loses the factor of 3. After you differentiate the outer function, always ask out loud what was inside and multiply by its derivative. The correct answer is $3 cos (3 x)$ .

Reversing the quotient rule numerator

Swapping the two products flips the sign of the entire derivative. Test your memory of the order on something you can verify: $x^{2} / x$ is just $x$ , so its derivative must be 1. The correct order gives $\frac{2 x \cdot x - x ^{2} \cdot 1}{x ^{2}} = 1$ , and the reversed order gives $- 1$ .

Multiplying the two derivatives

The derivative of a product is not the product of the derivatives. With $f (x) = g (x) = x$ , the product is $x^{2}$ and its derivative is $2 x$ , but multiplying the derivatives gives 1. The same warning applies to quotients and to compositions.

Practice

Answers are checked here — nothing is sent anywhere.

1
$\frac{d}{d x} (7 x^{5} - 3 x^{2})$
2
$\frac{d}{d x} ((2 x + 1)^{4})$
3
$\frac{d}{d x} (\frac{x ^{2}}{x + 1})$

Frequently asked questions

Does it tell me which rule it used at each step?

Yes. Every line names the rule being applied — power, product, quotient, or chain — so you can find the exact line where your own working diverged instead of comparing final answers.

Can it do implicit differentiation?

Yes. Ask for dy/dx from an equation such as x² + y² = 25. The working differentiates both sides with respect to x, attaches dy/dx to every term containing y, then isolates it, giving dy/dx = −x/y.

How do I get a second or third derivative?

Ask for it directly, for example "find the second derivative of x⁴ − 3x²". Each round of differentiation is shown separately so you can see the degree drop, which is the usual check that nothing was lost.

My answer looks different from the one shown. Am I wrong?

Often not. Derivatives can be written in several equivalent forms, especially after a quotient rule. Substitute a convenient number such as x = 2 into both expressions; if they agree, they are the same function.

What does it cost?

The first solution is free and needs no account. A free account gives three solutions a day. Gauth Plus removes the limit for $11.99 a month with a three-day free trial.

Keep going

Calculus solverDerivatives, integrals, limits, and series worked line by line in one place.The chain rule, explained plainlyHow to spot a composite function and peel it apart without losing a factor.The product and quotient rulesWhy the quotient rule looks the way it does, and the sign error to avoid.Integral calculatorRun differentiation backwards and check any antiderivative you produce.