Differentiation

1. Basic Definitions

Derivative using First Principle
The derivative of $f(x)$ with respect to $x$, at $x=a$ is given by:

$$ f'(a) = \lim_{h\rightarrow 0} \left[ \frac{f(a+h)-f(a)}{h} \right] $$

The derivative for $f(x)$ at any general point $x$ is:

$$ f'(x) = \lim_{h\rightarrow 0} \left[ \frac{f(x+h)-f(x)}{h} \right] $$
Notation & Small Increments
If $y=f(x)$, its derivative is written as $\frac{dy}{dx} = f'(x)$.

For a small increment $\delta x$ in $x$ and corresponding increment $\delta y$ in $y$:

$$ \lim_{\delta x\rightarrow 0} \left( \frac{\delta y}{\delta x} \right) = \frac{dy}{dx} $$

$y=f(x)$	$\frac{dy}{dx}=f'(x)$	$y=f(x)$	$\frac{dy}{dx}=f'(x)$
$c$ (Constant)	$0$	$\sec x$	$\sec x \tan x$
$x^n$	$n x^{n-1}$	$\text{cosec } x$	$-\text{cosec } x \cot x$
$\frac{1}{x}$	$-\frac{1}{x^2}$	$\cot x$	$-\text{cosec}^2 x$
$\frac{1}{x^n}$	$-\frac{n}{x^{n+1}}$	$e^x$	$e^x$
$\sqrt{x}$	$\frac{1}{2\sqrt{x}}$	$a^x$	$a^x \log a$
$\sin x$	$\cos x$	$\log x$	$\frac{1}{x}$
$\cos x$	$-\sin x$	$\log_a x$	$\frac{1}{x \log a}$
$\tan x$	$\sec^2 x$

If $u$ and $v$ are differentiable functions of $x$:

Addition & Subtraction Rule:
If $y = u \pm v$, then $\frac{dy}{dx} = \frac{du}{dx} \pm \frac{dv}{dx}$
Product Rule:
If $y = uv$, then $\frac{dy}{dx} = u\frac{dv}{dx} + v\frac{du}{dx}$
Quotient Rule:
If $y = \frac{u}{v}$ (where $v \neq 0$), then

$$ \frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2} $$

The Chain Rule
If $y=f(u)$ and $u=g(x)$, then:

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

Alternatively: $\frac{dy}{dx} = f'[g(x)] \cdot g'(x)$

For three variables: $\frac{dy}{dx} = \frac{dy}{dv} \times \frac{dv}{du} \times \frac{du}{dx}$

$y$	$\frac{dy}{dx}$
$[f(x)]^n$	$n[f(x)]^{n-1} \cdot f'(x)$
$\sqrt{f(x)}$	$\frac{f'(x)}{2\sqrt{f(x)}}$
$\frac{1}{[f(x)]^n}$	$\frac{-n \cdot f'(x)}{[f(x)]^{n+1}}$
$\sin[f(x)]$	$\cos[f(x)] \cdot f'(x)$
$\cos[f(x)]$	$-\sin[f(x)] \cdot f'(x)$
$\tan[f(x)]$	$\sec^2[f(x)] \cdot f'(x)$
$\sec[f(x)]$	$\sec[f(x)] \cdot \tan[f(x)] \cdot f'(x)$
$\text{cosec}[f(x)]$	$-\text{cosec}[f(x)] \cdot \cot[f(x)] \cdot f'(x)$
$a^{f(x)}$	$a^{f(x)} \cdot \log a \cdot f'(x)$
$e^{f(x)}$	$e^{f(x)} \cdot f'(x)$
$\log[f(x)]$	$\frac{f'(x)}{f(x)}$
$\log_a[f(x)]$	$\frac{f'(x)}{f(x)\log a}$

Interpretation of Derivative
The derivative of a function $y=f(x)$ at any point $P(x_1, y_1)$ is the slope of the tangent at that point on the curve.

Slope of Secant PQ: $\frac{f(a+h)-f(a)}{h}$

Slope of Tangent at P: $f'(a) = \lim_{h \rightarrow 0} \left[ \frac{f(a+h)-f(a)}{h} \right]$

Geometrical meaning of derivative (Secants approaching Tangent)

Derivative of Inverse Function
If $y=f(x)$ is a differentiable function and $x=f^{-1}(y)$ exists, then:

$$ \frac{dx}{dy} = \frac{1}{dy/dx} \quad \left(\text{where } \frac{dy}{dx} \neq 0\right) $$

Alternatively: $(f^{-1})'[f(x)] = \frac{1}{f'(x)}$

Graph of inverse secant function $y = \sec^{-1} x$

Important Formulae for Inverse Trig
- $\sin^{-1}(\sin \theta) = \theta$, \ $\sin(\sin^{-1} x) = x$
- $\cos^{-1}(\cos \theta) = \theta$, \ $\cos(\cos^{-1} x) = x$
- $\tan^{-1}(\tan \theta) = \theta$, \ $\tan(\tan^{-1} x) = x$
- $\sin^{-1}(\cos \theta) = \sin^{-1}[\sin(\frac{\pi}{2} - \theta)] = \frac{\pi}{2} - \theta$
- $\cos^{-1}(\sin \theta) = \cos^{-1}[\cos(\frac{\pi}{2} - \theta)] = \frac{\pi}{2} - \theta$
- $\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}$
- $\tan^{-1} x + \cot^{-1} x = \frac{\pi}{2}$
- $\sec^{-1} x + \text{cosec}^{-1} x = \frac{\pi}{2}$
- $\tan^{-1} x \pm \tan^{-1} y = \tan^{-1}\left(\frac{x \pm y}{1 \mp xy}\right)$

Expression	Substitutions
$\sqrt{1-x^2}$	$x = \sin \theta \text{ or } x = \cos \theta$
$\sqrt{1+x^2}$	$x = \tan \theta \text{ or } x = \cot \theta$
$\sqrt{x^2-1}$	$x = \sec \theta \text{ or } x = \text{cosec } \theta$
$\sqrt{\frac{a+x}{a-x}} \text{ or } \sqrt{\frac{a-x}{a+x}}$	$x = a\cos 2\theta \text{ or } x = a\cos \theta$
$\sqrt{\frac{1+x}{1-x}} \text{ or } \sqrt{\frac{1-x}{1+x}}$	$x = \cos 2\theta \text{ or } x = \cos \theta$
$\frac{2x}{1+x^2}, \frac{1-x^2}{1+x^2}, \frac{2x}{1-x^2}$	$x = \tan \theta$
$3x - 4x^3$	$x = \sin \theta$
$4x^3 - 3x \text{ or } 2x^2 - 1$	$x = \cos \theta$
$\frac{3x - x^3}{1 - 3x^2}$	$x = \tan \theta$

Logarithmic Differentiation
To differentiate $y = [f(x)]^{g(x)}$, take logarithms on both sides:

$\log y = g(x) \log[f(x)]$

Shortcut Form: $\frac{d}{dx}f^g = f^g \left[ \frac{g}{f} \cdot f' + (\log f) \cdot g' \right]$
Implicit Functions Property
An implicit function of the form $x^m y^n = (x+y)^{m+n}$, where $m, n \in R$, always has a first-order derivative $\frac{dy}{dx} = \frac{y}{x}$ and a second-order derivative $\frac{d^2y}{dx^2} = 0$.

Parametric Functions
If $x=f(t)$ and $y=g(t)$ are differentiable functions of parameter $t$, then:

$$ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \quad \left(\text{where } \frac{dx}{dt} \neq 0\right) $$

Derivative of one function w.r.t another: If $u=f(x)$ and $v=g(x)$, then $\frac{du}{dv} = \frac{du/dx}{dv/dx} = \frac{f'(x)}{g'(x)}$
Higher Order Derivatives
The $n^{\text{th}}$ order derivative is denoted by $y^{(n)} = f^{(n)}(x) = \frac{d^ny}{dx^n}$.
- $y = x^m \implies y_n = \frac{m!}{(m-n)!}x^{m-n}$ \ (if $m>0, m>n$)
- $y = \frac{1}{ax+b} \implies y_n = \frac{(-1)^n n! a^n}{(ax+b)^{n+1}}$
- $y = \log x \implies y_n = \frac{(-1)^{n-1}(n-1)!}{x^n}$
- $y = \sin x \implies y_n = \sin\left(\frac{n\pi}{2} + x\right)$
- $y = \cos(ax+b) \implies y_n = a^n \cos\left(\frac{n\pi}{2} + ax + b\right)$
- $y = e^{ax}\sin(bx+c) \implies y_n = e^{ax}(a^2+b^2)^{n/2} \sin(bx+c+n\alpha)$ where $\alpha = \tan^{-1}\left(\frac{b}{a}\right)$