Derivative Rules Step-by-Step

Pick any function from the library and watch the full differentiation unfold step by step. Every rule is labeled, every transition explained, with formulas rendered in LaTeX.

Rule Category

Select a Function

DifferentiatingPower Rule

1Identify the power rule patternPower Rule

The function has the form x^n where n is a constant.

2Apply the power rule: bring down the exponent, reduce by 1Power Rule

3Simplify the exponentPower Rule

Final Answer

Rule Reference: Power Rule

Multiply by the exponent, then reduce the exponent by 1. Works for any real number n.

Reference Guide

The Six Core Differentiation Rules

Power rule

\dfrac{d}{dx} x^n = n x^{n-1}

Constant rule

\dfrac{d}{dx} c = 0

Sum / Difference rule

\dfrac{d}{dx}[f \pm g] = f' \pm g'

Product rule

\dfrac{d}{dx}[uv] = u'v + uv'

Quotient rule

\dfrac{d}{dx}\left[\frac{u}{v}\right] = \frac{u'v - uv'}{v^2}

Chain rule

\dfrac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

Chain Rule Strategy

The chain rule applies whenever you differentiate a composition $f(g(x))$ .

Identify the outer and inner functions. For $\sin(x^2)$ , the outer is $\sin(u)$ and the inner is $u = x^2$ .
Differentiate the outer function, leaving the inner function unchanged inside.
Multiply by the derivative of the inner function.

Example

\dfrac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot 2x = 2x\cos(x^2)

For nested compositions, apply the chain rule repeatedly, working from the outermost layer inward.

Product vs Quotient Rule

Both rules handle functions made by combining two differentiable pieces.

Product rule for

f(x) = u(x) \cdot v(x)

f' = u'v + uv'

Mnemonic: derivative of the first times the second, plus the first times derivative of the second.

Quotient rule for

f(x) = u(x)/v(x)

f' = \dfrac{u'v - uv'}{v^2}

Mnemonic: low d-high minus high d-low, all over low squared. The denominator is always $v^2$ , never $v$ alone.

Avoid the quotient rule when possible: rewriting $1/x^n$ as $x^{-n}$ lets you use the simpler power rule instead.

Trig and Exponential Quick Reference

f(x)	f'(x)
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\sec^2 x$
$e^x$	$e^x$
$\ln x$	$1/x$
$a^x$	$a^x \ln a$

These standard derivatives are derived from first principles and should be memorized. Combined with the chain rule, they handle all compositions of these forms.

Related Content