Derivative Calculator

Every rule, explained.

Explore the rules the calculator uses to solve derivatives—from constants and powers to products, quotients, chains, and trigonometric functions. Each rule includes a clear explanation, when to use it, and worked examples to help you understand the steps.

Constant rule

\[\frac{d}{dx}c = 0\]

A quantity that never changes has no rate of change.

Variable rule

\[\frac{d}{dx}x = 1\]

The variable changes exactly as fast as itself.

Sum rule

\[\left(f + g\right)' = f' + g'\]

Differentiate term by term, then add the results back together.

Constant multiple rule

\[\left(cf\right)' = c\,f'\]

A constant factor rides along untouched while the function is differentiated.

Power rule

\[\frac{d}{dx}x^{n} = n x^{n-1}\]

Bring the exponent down as a factor, then reduce the exponent by one.

Product rule

\[\left(fg\right)' = f'g + fg'\]

Differentiate each factor in turn, leaving the other one alone.

Quotient rule

\[\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^{2}}\]

Bottom times the derivative of the top, minus top times the derivative of the bottom, all over the bottom squared.

Chain rule

\[\left(f \circ g\right)' = f'(g)\,g'\]

Differentiate the outside function, keep the inside unchanged, then multiply by the derivative of the inside.

Exponential rule

\[\frac{d}{dx}a^{x} = a^{x}\ln a\]

An exponential grows in proportion to its own size; the constant of proportionality is \(\ln a\).

Logarithm rule

\[\frac{d}{dx}\ln x = \frac{1}{x}\]

The natural logarithm differentiates to the reciprocal of its argument.

Logarithm rule, base a

\[\frac{d}{dx}\log_{a} x = \frac{1}{x\ln a}\]

A logarithm to any constant base is the natural logarithm scaled by a fixed factor.

Trigonometric rule

\[\frac{d}{dx}\sin x = \cos x\]

The six trigonometric functions differentiate into each other in a short, closed cycle.

Inverse trigonometric rule

\[\frac{d}{dx}\arctan x = \frac{1}{1 + x^{2}}\]

The inverse trigonometric functions differentiate to algebraic expressions — no trigonometry left in the answer.

Hyperbolic rule

\[\frac{d}{dx}\tanh x = \operatorname{sech}^{2} x\]

The hyperbolic functions behave like the trigonometric ones, but without the alternating minus signs.

Inverse hyperbolic rule

\[\frac{d}{dx}\operatorname{artanh} x = \frac{1}{1 - x^{2}}\]

Like the inverse trigonometric derivatives, but with the signs inside the roots flipped.

Absolute value rule

\[\frac{d}{dx}\left|x\right| = \operatorname{sgn} x, \; x \neq 0\]

Away from the corner, the absolute value is a straight line of slope \(\pm 1\).

Logarithmic differentiation

\[\frac{d}{dx}f^{g} = f^{g}\left(g'\ln f + \frac{gf'}{f}\right)\]

When the variable is in the base and the exponent at once, take logarithms first.

Rewrite in natural logarithms

\[\log_{a} b = \frac{\ln b}{\ln a}\]

When the base of a logarithm contains the variable, change to base \(e\) and use the quotient rule.