Derivative Rules: Complete Table with Visual Proofs

Q: What are the basic derivative rules?

Constant rule (c' = 0), power rule ((x^n)' = nx^{n-1}), constant multiple rule, sum and difference rule, product rule, quotient rule and chain rule, plus the derivatives of e^x, \ln x and the trig functions.

Q: When do I use the chain rule?

Whenever one function is plugged into another, like \sin(3x), (x^2+1)^5 or e^{-x^2}. Differentiate the outer function, keep the inside, then multiply by the derivative of the inside.

Q: Is the product rule just multiplying the derivatives?

No, and this is the most common mistake. (uv)' \neq u'v'. The explainer on this page shows why the correct formula is u'v + uv'.

The core rules

Rule	Formula	Note
Constant	$\frac{d}{dx}[c] = 0$	A flat line has slope 0.
Power	$\frac{d}{dx}[x^n] = n x^{n-1}$	Works for any real n.
Constant multiple	$\frac{d}{dx}[c\,f] = c\,f'$	Stretching a graph stretches its slopes.
Sum / difference	$(f \pm g)' = f' \pm g'$	Differentiate term by term.
Product	$(uv)' = u'v + uv'$	Not u′v′. See the explainer below.
Quotient	$\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$	Order matters in the numerator.
Chain	$\frac{d}{dx} f(g(x)) = f'(g(x))\,g'(x)$	Outer derivative times inner derivative.

Why the product rule has two terms

Think of $u \cdot v$ as the area of a rectangle with sides $u$ and $v$ . If both sides grow a little, the area grows by two strips and a tiny corner:

Grow u by du = 0.80Grow v by dv = 0.60

New area − old area = v·du 1.6 + u·dv 1.8 + du·dv 0.48

Shrink both sliders: the grey corner du·dv vanishes far faster than the two strips. What survives is $d(uv) = u\,dv + v\,du$ , the product rule.

The corner is a small change times a small change, so it vanishes compared with the strips as the changes shrink. Dividing the rest by $dx$ gives $(uv)' = u'v + uv'$ . The quotient rule follows by applying the product and chain rules to $u \cdot v^{-1}$ .

Derivatives of common functions

$f(x)$	$f'(x)$
$e^x$	$e^x$
$a^x$	$a^x \ln a$
$\ln x$	$\frac{1}{x}$
$\log_a x$	$\frac{1}{x \ln a}$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\sec^2 x$
$\sec x$	$\sec x \tan x$
$\csc x$	$-\csc x \cot x$
$\cot x$	$-\csc^2 x$
$\arcsin x$	$\frac{1}{\sqrt{1-x^2}}$
$\arccos x$	$-\frac{1}{\sqrt{1-x^2}}$
$\arctan x$	$\frac{1}{1+x^2}$
$\sinh x$	$\cosh x$
$\cosh x$	$\sinh x$

Each of these has its own page with the derivation and graph: e.g. ln x, sin x, tan x, or browse all of them.

Putting the rules together

Differentiate $x^2 e^{3x}$ : it is a product, and the second factor needs the chain rule. $(x^2)' e^{3x} + x^2 (e^{3x})' = 2x e^{3x} + 3x^2 e^{3x}$ . The derivative calculator shows this kind of breakdown for any function you type.

Questions students ask

What are the basic derivative rules?

Constant rule ( $c' = 0$ ), power rule ( $(x^n)' = nx^{n-1}$ ), constant multiple rule, sum and difference rule, product rule, quotient rule and chain rule, plus the derivatives of $e^x$ , $\ln x$ and the trig functions.

How do I remember the quotient rule?

“Low d-high minus high d-low, over the square of what’s below”: $\left(\frac{u}{v}\right)' = \frac{v\,u' - u\,v'}{v^2}$ . The order in the numerator matters because of the minus sign.

When do I use the chain rule?

Whenever one function is plugged into another, like $\sin(3x)$ , $(x^2+1)^5$ or $e^{-x^2}$ . Differentiate the outer function, keep the inside, then multiply by the derivative of the inside.

Is the product rule just multiplying the derivatives?

No, and this is the most common mistake. $(uv)' \neq u'v'$ . The explainer on this page shows why the correct formula is $u'v + uv'$ .

The core rules

Why the product rule has two terms

Derivatives of common functions

Putting the rules together

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Questions students ask