What is the most important derivative rule?

The power rule is used most often, but the chain rule matters most in practice: nearly every function beyond the basics has an inside function, and forgetting the chain rule is the most common derivative mistake.

Does the power rule work for negative and fractional exponents?

Yes. (d)/(dx)x^-2=-2x^-3 and (d)/(dx)x^1/2=frac12x^-1/2. Rewrite roots and fractions as powers of x first.

Is the derivative of a product the product of the derivatives?

No. For example, (d)/(dx)(x· x)=(d)/(dx)x^2=2x, but the product of the derivatives is 1·1=1. Use the product rule (fg)'=f'g+fg'.

What is the derivative of a constant?

Zero. A constant function doesn't change, so its rate of change is 0 everywhere; its graph is a horizontal line with slope 0.

What is the derivative of e^x?

e^x itself. It is the only exponential (apart from multiples of it) whose rate of change equals its value at every point.

Do I need to memorize all the derivative rules?

Most classes expect you to know the power, product, quotient and chain rules and the derivatives of e^x, ln x, sin x and cos x by heart. The others can be worked out from these, though it's quicker to know them.

Derivative Rules: The Complete List, With Examples

Why do we need derivative rules?

The definition of the derivative, $f^{'} (x) =$ $lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$ , works for any function, but every use means expanding, simplifying and taking a limit. The rules below are shortcuts: each one is that limit worked out once, in general, so you never have to repeat it. Together they let you differentiate any function built from powers, exponentials, logarithms and trig functions, combined in any way.

What is the complete list of derivative rules?

In the table, $c$ and $n$ are fixed numbers, and $f$ and $g$ are functions of $x$ .

Function	Derivative	Rule
$c$	$0$	constant rule
$x^{n}$	$n x^{n - 1}$	power rule (any number $n$ )
$c f (x)$	$c f^{'} (x)$	constant multiple rule
$f (x) \pm g (x)$	$f^{'} (x) \pm g^{'} (x)$	sum and difference rules
$f (x) g (x)$	$f^{'} (x) g (x) + f (x) g^{'} (x)$	product rule
$\frac{f (x)}{g (x)}$	$\frac{f^{'} (x) g (x) - f (x) g^{'} (x)}{g (x)^{2}}$	quotient rule
$f (g (x))$	$f^{'} (g (x)) g^{'} (x)$	chain rule
$e^{x}$	$e^{x}$	natural exponential
$b^{x}$	$b^{x} \ln b$	other exponentials
$\ln x$	$\frac{1}{x}$	natural logarithm ( $x > 0$ )
$\log_{b} x$	$\frac{1}{x \ln b}$	other logarithms
$\sin x$ , $\cos x$	$\cos x$ , $- \sin x$	sine and cosine
$\tan x$ , $\sec x$	$\sec^{2} x$ , $\sec x \tan x$	tangent and secant
$\cot x$ , $\csc x$	$- \csc^{2} x$ , $- \csc x \cot x$	cotangent and cosecant
$\arcsin x$ , $\arccos x$	$\frac{1}{\sqrt{1 - x^{2}}}$ , $- \frac{1}{\sqrt{1 - x^{2}}}$	inverse sine and cosine
$\arctan x$	$\frac{1}{1 + x^{2}}$	inverse tangent

Tip

Trig derivatives only take these forms when angles are measured in radians, the standard unit in calculus.

How do you use the power rule?

The power rule says: to differentiate a power of $x$ , bring the exponent down in front and subtract $1$ from the exponent.

The power rule

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

n x^{n - 1}

for any number

n

Why it works

For a whole number $n$ , multiply out $(x + h)^{n} =$ $x^{n} + n x^{n - 1} h + (terms with h^{2} or higher)$ .
The difference quotient is $\frac{(x + h)^{n} - x^{n}}{h} =$ $n x^{n - 1} + (terms with at least one h)$ .
As $h \to 0$ , every term with an $h$ goes to $0$ , leaving $n x^{n - 1}$ . (Other exponents are covered later in the course, with logarithms.)

The power rule only applies to $x^{n}$ , so roots and fractions must first be rewritten as powers: $\sqrt{x} = x^{1 / 2}$ , $\sqrt[3]{x^{2}} =$ $x^{2 / 3}$ , and $\frac{1}{x^{n}} = x^{- n}$ .

Worked example: Rewrite, then use the power rule

Differentiate $y =$ $4 x^{5} - \frac{3}{x^{2}} + 2 \sqrt{x}$ .

Rewrite every term as a power of $x$ : $y =$ $4 x^{5} - 3 x^{- 2} + 2 x^{1 / 2}$ .
Differentiate term by term (sum and constant multiple rules): $4 \cdot 5 x^{4} - 3 \cdot (- 2) x^{- 3} + 2 \cdot \frac{1}{2} x^{- 1 / 2}$ .
Simplify: $20 x^{4} + 6 x^{- 3} + x^{- 1 / 2}$ .
Rewrite without negative or fractional exponents if you like: $20 x^{4} + \frac{6}{x^{3}} + \frac{1}{\sqrt{x}}$ .

Answer: $y^{'} =$ $20 x^{4} + \frac{6}{x^{3}} + \frac{1}{\sqrt{x}}$

Common mistake

Using the power rule on an exponential. $\frac{d}{d x} 2^{x}$ is not $x \cdot 2^{x - 1}$ . The power rule needs the variable in the base and a fixed exponent; $2^{x}$ has a fixed base and the variable in the exponent, so it uses $\frac{d}{d x} b^{x} =$ $b^{x} \ln b$ , giving $2^{x} \ln 2$ .

What are the sum, difference and constant multiple rules?

The derivative of a sum is the sum of the derivatives, and a number multiplying a function just stays put. So you can split a long expression into its terms and differentiate each one on its own, as in the example above. This only works for adding, subtracting and multiplying by a number; multiplying two functions together needs the product rule.

What are the product and quotient rules?

The derivative of a product is not the product of the derivatives. The product rule differentiates one factor at a time while leaving the other alone, and adds the results. The quotient rule is the rule for one function divided by another; the order on top matters because of the minus sign.

Worked example: The product rule

Differentiate $y = x^{3} \sin x$ .

Name the factors: $f = x^{3}$ with $f^{'} = 3 x^{2}$ , and $g = \sin x$ with $g^{'} = \cos x$ .
Product rule: $y^{'} =$ $f^{'} g + f g^{'} =$ $3 x^{2} \sin x + x^{3} \cos x$ .

Answer: $y^{'} =$ $3 x^{2} \sin x + x^{3} \cos x$

Worked example: The quotient rule

Differentiate $y = \frac{e^{x}}{x + 1}$ .

Name the top $f = e^{x}$ with $f^{'} = e^{x}$ , and the bottom $g = x + 1$ with $g^{'} = 1$ .
Quotient rule: $y^{'} =$ $\frac{f^{'} g - f g^{'}}{g^{2}} =$ $\frac{e^{x} (x + 1) - e^{x} \cdot 1}{(x + 1)^{2}}$ .
Simplify the top: $e^{x} (x + 1) - e^{x} =$ $x e^{x} + e^{x} - e^{x} =$ $x e^{x}$ .

Answer: $y^{'} =$ $\frac{x e^{x}}{(x + 1)^{2}}$

Both rules, why they work, and when a rewrite is quicker than the quotient rule are covered in the product and quotient rules.

What is the chain rule?

The chain rule handles a function applied to something other than plain $x$ , such as $\sin (x^{2})$ or $(3 x + 1)^{5}$ . Differentiate the outside function, leaving the inside alone, then multiply by the derivative of the inside.

Worked example: The chain rule

Differentiate $y = \sin (x^{2})$ .

The inside is $x^{2}$ , with derivative $2 x$ . The outside is $\sin ()$ , with derivative $\cos ()$ .
Outside's derivative with the inside unchanged: $\cos (x^{2})$ .
Multiply by the inside's derivative: $y^{'} =$ $\cos (x^{2}) \cdot 2 x =$ $2 x \cos (x^{2})$ .

Answer: $y^{'} = 2 x \cos (x^{2})$

See the chain rule for longer chains and the most common mistakes, and derivatives of $e^{x}$ and $\ln x$ for exponentials and logarithms.

How do you know which derivative rule to use?

Look at how the function is built and find the last operation you would do to compute it for a particular $x$ . That operation chooses the first rule; then use the rules again on each piece until only basic functions from the table are left.

The last operation is	Example	Use first
adding or subtracting terms	$x^{4} - 3 \sin x$	sum rule: term by term
multiplying two pieces that contain $x$	$x^{2} e^{x}$	product rule
dividing by a piece that contains $x$	$\frac{\sin x}{x^{2} + 1}$	quotient rule
applying a function to something other than $x$	$\sqrt{x^{2} + 1}$ , $e^{3 x}$	chain rule
nothing: a basic function of $x$	$\tan x$ , $\ln x$	the formula from the table

Tip

Before choosing, check whether a rewrite makes things simpler: $\frac{x^{3} + 2 x}{x} =$ $x^{2} + 2$ needs no quotient rule, and $\ln (x^{2}) = 2 \ln x$ needs no chain rule. The course's Playbook walks through this choice question by question.

Derivative rules: the complete list, with examples