Rules of differentiation

A handful of rules, applied repeatedly, differentiate every elementary function. We list them here, prove the chain rule (the most powerful), and assemble a table of standard derivatives. Subsequent subtopics extend the rules to implicit, logarithmic, and parametric forms.

The four basic rules

For differentiable $f, g$:

1. Sum rule: $(f \pm g)'(x) = f'(x) \pm g'(x)$.
2. Constant multiple: $(cf)'(x) = c f'(x)$ for constant $c$.
3. Product rule: $(fg)'(x) = f'(x) g(x) + f(x) g'(x)$.
4. Quotient rule: $\left(\dfrac{f}{g}\right)'(x) = \dfrac{f'(x) g(x) - f(x) g'(x)}{g(x)^2}$, provided $g(x) \neq 0$.

The sum rule and constant multiple are immediate from the limit definition. The product rule requires a small trick (add and subtract $f(x + h) g(x)$). The quotient rule follows from product + chain (differentiate $f \cdot (1/g)$).

The chain rule

For differentiable $f$ and $g$, with $f \circ g$ defined:

$(f \circ g)'(x) = f'(g(x)) \cdot g'(x).$

In Leibniz notation: $\dfrac{dy}{dx} = \dfrac{dy}{du}\cdot \dfrac{du}{dx}$, where $u = g(x)$ and $y = f(u)$.

Proof sketch. $\dfrac{f(g(x + h)) - f(g(x))}{h} = \dfrac{f(g(x + h)) - f(g(x))}{g(x + h) - g(x)} \cdot \dfrac{g(x + h) - g(x)}{h}$. As $h \to 0$, $g(x + h) \to g(x)$ (continuity); the first factor $\to f'(g(x))$, the second $\to g'(x)$.

(A careful proof needs to handle the case $g(x + h) = g(x)$ separately.)

Table of standard derivatives

Combining the rules

Most school problems reduce to applying these in combination. Identify the outermost operation first, apply the corresponding rule, and chain through.

For $f(x) = \sin(x^3)$: outermost is $\sin$. Chain rule: $f'(x) = \cos(x^3) \cdot 3x^2$.

For $f(x) = \dfrac{x^2}{1 + x}$: quotient rule. $f'(x) = \dfrac{2x(1 + x) - x^2}{(1 + x)^2} = \dfrac{x^2 + 2x}{(1 + x)^2}$.

For $f(x) = e^{x} \sin x$: product. $f'(x) = e^x \sin x + e^x \cos x = e^x(\sin x + \cos x)$.

Worked examples

Example 1. Differentiate $f(x) = (3x + 1)^5$.

Chain: $f'(x) = 5(3x + 1)^4 \cdot 3 = 15(3x + 1)^4$.

Example 2. Differentiate $f(x) = \sin(\log x)$.

Chain: $f'(x) = \cos(\log x) \cdot \dfrac{1}{x}$.

Example 3. Differentiate $f(x) = e^{x^2 + 1}$.

Chain: $f'(x) = e^{x^2 + 1} \cdot 2x$.

Example 4. Differentiate $f(x) = \dfrac{\sin x}{1 + \cos x}$.

Quotient: $f'(x) = \dfrac{\cos x (1 + \cos x) - \sin x (-\sin x)}{(1 + \cos x)^2} = \dfrac{\cos x + \cos^2 x + \sin^2 x}{(1 + \cos x)^2} = \dfrac{\cos x + 1}{(1 + \cos x)^2} = \dfrac{1}{1 + \cos x}$.

Example 5. Differentiate $f(x) = \tan^{-1}\left(\dfrac{2x}{1 - x^2}\right)$.

Use the identity $\tan^{-1}\left(\dfrac{2x}{1 - x^2}\right) = 2 \tan^{-1} x$ for $|x| < 1$. So $f'(x) = \dfrac{2}{1 + x^2}$. (Outside that range a $\pm \pi$ adjustment changes nothing for the derivative.)

Example 6. Differentiate $f(x) = \cos^2(\sqrt{x})$.

Outer: square. Inner: $\cos$ of $\sqrt{x}$. $f'(x) = 2\cos(\sqrt{x}) \cdot (-\sin\sqrt{x}) \cdot \dfrac{1}{2\sqrt{x}} = -\dfrac{\sin(2\sqrt{x})}{2\sqrt{x}}$ (using $2 \sin \theta \cos \theta = \sin 2\theta$).

Try it yourself

1. Differentiate $f(x) = x^7 + 5x^3 - 2x$.
2. Differentiate $f(x) = \sin x \cos x$ using both product rule and double-angle identity.
3. Differentiate $f(x) = \dfrac{1}{x^2 + 1}$.
4. Differentiate $f(x) = \tan(x^2)$.
5. Differentiate $f(x) = e^{3x} \cos(2x)$.
6. Differentiate $f(x) = \ln(\sin x)$.
7. Differentiate $f(x) = \sqrt{1 + x^2}$.
8. Differentiate $f(x) = (1 + x)(1 + x^2)(1 + x^3)$.
9. Differentiate $f(x) = \dfrac{x + 1}{x - 1}$.
10. Differentiate $f(x) = \sin^{-1}(2x \sqrt{1 - x^2})$ for $|x| < 1/\sqrt 2$. (Hint: $= 2 \sin^{-1} x$.)
11. Differentiate $f(x) = x e^{-x^2}$.
12. Differentiate $f(x) = \log(\cos(x^2))$.
13. Differentiate $f(x) = (\sin x)^3 \cdot \cos x$ , use product rule.
14. Differentiate $f(x) = \dfrac{\tan x}{1 + \tan^2 x}$, recognising it equals $\sin x \cos x$.

Pitfalls / Tricks

• Chain rule first, simplify later. Many students try to simplify before differentiating and lose track.
• For quotient rule, be careful with the sign in the numerator: $f' g - f g'$, not $f g' - f' g$.
• Trig identities can sometimes simplify before differentiating , but only do so when it's obvious.
• $\dfrac{d}{dx}(a^x) = a^x \ln a$, not $x a^{x - 1}$. Mixing power-rule and exp-rule is a frequent error.
• For inverse trig derivatives, watch the domain , the formula gives a real number only inside the natural domain.

Next, implicit and logarithmic differentiation.