Standard Rules of Differentiation

Differentiating every function from the limit definition would be slow and error-prone. A small set of standard rules of differentiation — each of them itself provable from first principles — lets almost any function built from simpler pieces be differentiated quickly.

Constant Rule

A constant does not change as $x$ changes, so its derivative is always zero:

Power Rule

For any real number $n$:

Example: $\dfrac{d}{dx}(x^6) = 6x^5$.

Constant Multiple Rule

A constant multiplying a function simply carries through the differentiation unchanged.

Sum and Difference Rule

A sum or difference of functions is differentiated term by term.

Product Rule

When $y = u(x)\cdot v(x)$ is a product of two functions:

In words: the first function times the derivative of the second, plus the second function times the derivative of the first. It is a genuine error to differentiate $u$ and $v$ separately and multiply the results — $\frac{d}{dx}(uv) \neq u'v'$ in general.

Quotient Rule

When $y = \dfrac{u(x)}{v(x)}$ (with $v(x)\neq 0$):

In words: denominator times derivative of numerator, minus numerator times derivative of denominator, all over the square of the denominator. Reversing the order of subtraction in the numerator flips the sign of the entire answer, so the order must always be kept exactly as stated.

Chain Rule (Function of a Function)

When $y$ is a function of $u$, and $u$ is itself a function of $x$ — that is, $y=f(u)$ where $u=g(x)$ — the chain rule states:

$\frac{dy}{dx} = \frac{dy}{du}\times\frac{du}{dx}$ …