Rules of Differentiation

Differentiating every function from first principles, using limits, would be slow and error-prone. Instead, a small set of standard rules of differentiation — each itself provable from first principles — lets us differentiate almost any function built from simpler pieces almost immediately.

Constant Rule

The derivative of a constant is always zero, because a constant does not change as $x$ changes:

Power Rule

For any real number $n$:

Example: $\dfrac{d}{dx}(x^5) = 5x^4$.

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.

Example (Power + Sum/Difference together): $\dfrac{d}{dx}(3x^4 - 5x^2 + 7x - 9) = 12x^3 - 10x + 7$ (the constant $-9$ differentiates to $0$).

Product Rule

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

In words: derivative of the first times the second, plus the first times the derivative of the second. 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 $y = \dfrac{u(x)}{v(x)}$ is a ratio of two functions (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. The order of subtraction in the numerator matters — reversing it flips the sign of the answer.

Chain Rule (Function of a Function)

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

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