Calculus Reference

Derivative Rules

Constant, power, sum, product, quotient, and chain rules cover the standard forms used to differentiate algebraic functions.

Derivative Rule Index

Formula summaries and worked examples for the standard differentiation rules.

Derivative Rules at a Glance

RuleDerivativeUse it when
Constant$\frac{d}{dx}[c]=0$The term has no variable.
Power$\frac{d}{dx}[x^n]=nx^{n-1}$The variable is raised to a constant power.
Constant multiple$(cf)'=cf'$A constant multiplies a function.
Sum or difference$(f\pm g)'=f'\pm g'$Terms are added or subtracted.
Product$(fg)'=f'g+fg'$Two variable expressions are multiplied.
Quotient$\left(\frac fg\right)'=\frac{f'g-fg'}{g^2}$One variable expression is divided by another.
Chain$\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$One function is inside another.

Constant Rule

A fixed value does not change as $x$ changes, so its derivative is zero.

Differentiate a Constant

Find the derivative of $f(x)=7$.

Solution Steps
  1. The expression has no variable term.
  2. Apply the constant rule.
$$f'(x)=\frac{d}{dx}[7]$$$$\boxed{f'(x)=0}$$

Power Rule and Exponents

Bring the exponent down as a coefficient, then subtract $1$ from the exponent. This works for positive, negative, and fractional constant exponents wherever the original function is defined.

Power Rule
$$\frac{d}{dx}[x^n]=nx^{n-1}$$
With a Coefficient
$$\frac{d}{dx}[ax^n]=anx^{n-1}$$

Positive Integer Exponent

Find the derivative of $f(x)=5x^3$.

Solution Steps
  1. Multiply the coefficient by the exponent.
  2. Reduce the exponent by one.
$$f'(x)=5(3)x^{3-1}$$$$\boxed{f'(x)=15x^2}$$

Fractional and Negative Exponents

Differentiate $g(x)=\sqrt{x}+\frac{3}{x^2}$ for $x>0$.

Solution Steps
  1. Rewrite the radical and denominator as powers: $g(x)=x^{1/2}+3x^{-2}$.
  2. Apply the power rule to both terms and simplify.
$$g'(x)=\frac12x^{-1/2}-6x^{-3}$$$$\boxed{g'(x)=\frac{1}{2\sqrt{x}}-\frac{6}{x^3}}$$

Constant Multiple Rule

Keep a constant factor outside and differentiate the function it multiplies.

Factor Outside a Polynomial

Find the derivative of $h(x)=4(x^3-2x)$.

Solution Steps
  1. Keep the factor $4$.
  2. Differentiate each term inside the parentheses.
$$h'(x)=4(3x^2-2)$$$$\boxed{h'(x)=12x^2-8}$$

Sum and Difference Rules

Differentiate each term separately. Keep each term's sign as you go.

Differentiate Term by Term

Find the derivative of $p(x)=x^4-3x^2+2x-7$.

Solution Steps
  1. Differentiate the four terms independently.
  2. Combine the resulting terms.
$$p'(x)=4x^3-6x+2+0$$$$\boxed{p'(x)=4x^3-6x+2}$$

Product Rule

For two factors, differentiate the first and keep the second, then keep the first and differentiate the second. The derivative is not usually the product of the two separate derivatives.

Product Rule
$$(fg)'=f'g+fg'$$

Two Factors

Find the derivative of $q(x)=x^2(x+3)$.

Solution Steps
  1. Let $f=x^2$ and $g=x+3$.
  2. Use $f'g+fg'$, then simplify.
$$q'(x)=(2x)(x+3)+x^2(1)$$$$\boxed{q'(x)=3x^2+6x}$$

Quotient Rule

For a quotient, use “bottom times derivative of top, minus top times derivative of bottom,” over the square of the bottom.

Quotient Rule
$$\left(\frac fg\right)'=\frac{f'g-fg'}{g^2},\quad g\ne0$$

Polynomial over a Linear Factor

Find the derivative of $r(x)=\frac{x^2+1}{x+1}$.

Solution Steps
  1. Identify the top as $f=x^2+1$ and the bottom as $g=x+1$.
  2. Apply the quotient rule and simplify the numerator.
$$r'(x)=\frac{2x(x+1)-(x^2+1)(1)}{(x+1)^2}$$$$\boxed{r'(x)=\frac{x^2+2x-1}{(x+1)^2}},\quad x\ne-1$$

Chain Rule

Use the chain rule when an expression is nested. Differentiate the outside function while keeping the inside, then multiply by the derivative of the inside.

Chain Rule
$$\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$$

Power of a Linear Expression

Find the derivative of $s(x)=(2x+1)^4$.

Solution Steps
  1. The outside function raises its input to the fourth power; the inside is $2x+1$.
  2. Differentiate the outside and multiply by the inside derivative, $2$.
$$s'(x)=4(2x+1)^3(2)$$$$\boxed{s'(x)=8(2x+1)^3}$$

Higher-Order Derivatives

The $n$th derivative is written $f^{(n)}(x)=\frac{d^n f}{dx^n}$. The notation $f^n$ usually means the function raised to a power. The derivative calculators show successive results through order $10$ for their listed function forms.

Repeated Power Rule

Find successive derivatives of $f(x)=x^4$.

Solution Steps
  1. Differentiate once to find $f'(x)$.
  2. Differentiate $f'(x)$ to find $f''(x)$.
  3. Differentiate again to find $f'''(x)$.
  4. Continue to the constant fourth derivative.
  5. Differentiate the constant to find the fifth derivative.
$$f'(x)=4x^3$$$$f''(x)=12x^2$$$$f'''(x)=24x$$$$f^{(4)}(x)=24$$$$f^{(5)}(x)=0$$