Differentiation Rules

Computing Derivatives Mechanically

Evaluating the limit definition each time is impractical. A handful of rules make differentiation routine algebra.

The basic rules:

• Power rule — $\frac{d}{dx}x^n = n\,x^{n-1}$: bring the exponent down, 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 rules for combining functions require care:

• Product rule — $(fg)' = f'g + fg'$. The derivative of a product is not the product of the derivatives; you add the two "differentiate one factor at a time" terms.
• Quotient rule — $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$. 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 the most error-prone.

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

Common pitfall: treating the derivative of a product as the product of derivatives, or dropping the inner factor in the chain rule. $(fg)' = f'g + fg'$, not $f'g'$. And for $f(g(x))$ you must multiply by $g'(x)$: $\frac{d}{dx}\sin(x^2) = \cos(x^2)\cdot 2x$, not just $\cos(x^2)$. Forgetting the inner derivative is the single most common differentiation error.