Differentiation Rules

Computing Derivatives Efficiently

Rather than evaluate the limit definition every time, a handful of rules compute derivatives mechanically. Mastering them turns differentiation into routine algebra.

The basic rules:

• Power rule — $\frac{d}{dx} x^n = n\, x^{n-1}$. Bring the exponent down and reduce it by one. So $\frac{d}{dx} x^3 = 3x^2$.
• Constant multiple — $\frac{d}{dx}(c\,f) = c\, f'$; constants factor out.
• Sum rule — the derivative of a sum is the sum of the derivatives.

The two rules for combining functions are where care is needed:

• Product rule — $(fg)' = f'g + f g'$. The derivative of a product is not the product of the derivatives; you must add the two "one factor differentiated at a time" terms.
• Quotient rule — $\left(\frac{f}{g}\right)' = \frac{f'g - f g'}{g^2}$. Note the minus sign and the order.
• Chain rule — for a composition $f(g(x))$, the derivative is $f'(g(x)) \cdot g'(x)$: differentiate the outer function (leaving the inner alone), then multiply by the derivative of the inner. This "outer times inner" rule is the most important and most error-prone.

Standard derivatives round out the toolkit: $\frac{d}{dx}\sin x = \cos x$, $\frac{d}{dx}e^x = e^x$ (its own derivative), and $\frac{d}{dx}\ln x = \frac{1}{x}$. With these plus the four combination rules, you can differentiate essentially any elementary function.

Common pitfall: thinking the derivative of a product is the product of the derivatives, or forgetting the chain rule on a composition. $(fg)' = f'g + fg'$, not $f'g'$. And for $f(g(x))$ you must multiply by $g'(x)$ — e.g. $\frac{d}{dx}\sin(x^2) = \cos(x^2)\cdot 2x$, not just $\cos(x^2)$. Omitting the inner derivative is the single most common differentiation mistake.