Derivatives and Differential Calculus Theorems

The derivative of $f$ at $x_0$ is defined as the limit of the difference quotient:

$$f'(x_0) = \lim_{h\to0}\frac{f(x_0+h)-f(x_0)}{h},$$

when this limit exists and is finite. Geometric interpretation: $f'(x_0)$ is the slope of the tangent line to the graph of $f$ at $(x_0,f(x_0))$. Physical interpretation: if $f$ is position, $f'$ is instantaneous velocity.

Left and right derivatives: if the limit is taken with $h\to0^-$ or $h\to0^+$ we obtain $f'_-(x_0)$ and $f'_+(x_0)$ respectively. $f$ is differentiable at $x_0$ iff both exist and are equal.

Differentiability implies continuity: if $f'(x_0)$ exists, then $f$ is continuous at $x_0$. The converse is false. Classic counterexample: $f(x)=|x|$ is continuous everywhere but not differentiable at $x=0$ because $f'_-(0)=-1\neq1=f'_+(0)$ (cusp).

Tangent line equation: $y = f(x_0) + f'(x_0)(x-x_0)$.

Fundamental derivatives (memorise):

• $(x^n)' = nx^{n-1}$ for any $n\in\mathbb{R}$ (including negative and fractional $n$)
• $(e^x)' = e^x$; $(a^x)' = a^x\ln a$
• $(\ln x)' = 1/x$; $(\log_a x)' = 1/(x\ln a)$
• $(\sin x)' = \cos x$; $(\cos x)' = -\sin x$; $(\tan x)' = 1/\cos^2 x = 1+\tan^2 x$
• $(\arcsin x)' = 1/\sqrt{1-x^2}$; $(\arccos x)' = -1/\sqrt{1-x^2}$; $(\arctan x)' = 1/(1+x^2)$
• $(\sinh x)' = \cosh x$; $(\cosh x)' = \sinh x$

Differentiation rules:

• Linearity: $(\alpha f + \beta g)' = \alpha f' + \beta g'$
• Product (Leibniz): $(fg)' = f'g + fg'$ — "derivative of first times second plus first times derivative of second"
• Quotient: $(f/g)' = (f'g - fg')/g^2$ (where $g\neq0$) — "derivative of numerator times denominator minus numerator times derivative of denominator, all over denominator squared"
• Chain rule: $[f(g(x))]' = f'(g(x))\cdot g'(x)$ — "derivative of outer function, evaluated at inner function, times derivative of inner function". Apply from outermost to innermost.
• Inverse function: if $g = f^{-1}$, then $g'(y) = 1/f'(g(y))$.

Rolle's theorem: if $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)$, then there exists $c\in(a,b)$ with $f'(c)=0$. Geometric interpretation: a smooth arc with equal endpoints has a horizontal tangent somewhere.

Lagrange's Mean Value Theorem: if $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists $c\in(a,b)$ such that:

$$f'(c) = \frac{f(b)-f(a)}{b-a}.$$

Geometric interpretation: there is a point where the tangent is parallel to the chord joining $(a,f(a))$ and $(b,f(b))$. This is one of the most important theorems in differential calculus.

Monotonicity corollaries (from Lagrange):

• $f'(x)>0$ on $(a,b)$ $\Rightarrow$ $f$ is strictly increasing on $[a,b]$
• $f'(x)<0$ on $(a,b)$ $\Rightarrow$ $f$ is strictly decreasing on $[a,b]$
• $f'(x)=0$ on $(a,b)$ $\Rightarrow$ $f$ is constant on $[a,b]$

Cauchy's generalised mean value theorem: if $f,g$ are continuous on $[a,b]$, differentiable on $(a,b)$, $g'\neq0$, then $\exists c\in(a,b)$ with $f'(c)/g'(c) = (f(b)-f(a))/(g(b)-g(a))$. This is the foundation for L'Hôpital's rule.

L'Hôpital's rule: let $f$ and $g$ be differentiable near $x_0$ (or at infinity), $g'\neq0$, and suppose $\lim f/g$ presents one of the indeterminate forms $0/0$ or $\infty/\infty$. Then:

$$\lim\frac{f}{g} = \lim\frac{f'}{g'},$$

provided the limit on the right exists (finite or $\pm\infty$). The rule may be applied repeatedly if the result is still indeterminate.

How to convert other forms to $0/0$ or $\infty/\infty$:

• $0\cdot\infty$: write $f\cdot g = f/(1/g)$ to get $0/0$
• $\infty-\infty$: find a common denominator or rationalise
• $0^0, 1^\infty, \infty^0$: apply the logarithm to convert to $0/0$ or $\infty/\infty$

Warning: the rule requires the limit of $f'/g'$ to exist. If $\lim f'/g'$ does not exist, the rule gives no information (it does not imply $\lim f/g$ fails to exist — it may still exist by other methods).

If $f$ is $n$ times differentiable at $x_0$, the Taylor polynomial of order $n$ centred at $x_0$ approximates $f$ near $x_0$:

$$f(x) = \sum_{k=0}^n\frac{f^{(k)}(x_0)}{k!}(x-x_0)^k + o((x-x_0)^n) \quad\text{as }x\to x_0.$$

The remainder $o((x-x_0)^n)$ (Peano form) is negligible compared to $(x-x_0)^n$ as $x\to x_0$.

Fundamental Maclaurin expansions ($x_0=0$):

• $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots = \displaystyle\sum_{k=0}^\infty \frac{x^k}{k!}$
• $\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots + o(x^{2n+1})$
• $\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots + o(x^{2n})$
• $\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots + o(x^n)$
• $(1+x)^\alpha = 1 + \alpha x + \frac{\alpha(\alpha-1)}{2}x^2 + \cdots + o(x^n)$ (binomial series)

Applications: computing limits of indeterminate forms (replace function by its Taylor expansion and simplify); approximating integrals; proving inequalities.