# Lecture 13: Differentation Rules

### The chain rule

﻿$f(x) = x^n$﻿ ﻿$f^1 (x) = n(x^{n-1}) * ( x^1 )$﻿

### The Product Rule

﻿$f(x) = xz$﻿ ﻿$f^1 (x) = x^1z + z^1x$﻿

### The Quotient Rule

﻿$f(x) = \frac{z}{x}$﻿ ﻿$f^1(x) = \frac{z^1x - zx^1}{x^2}$﻿

### The Derivative of exponentials/logarithmic functions

﻿$f(x) = log_ax$﻿ ﻿$f^1 (x) = \frac{1}{xIna} * (x^1)$﻿

﻿$f(x) = e^x$﻿ ﻿$f^1(x) = e^x$﻿

﻿$f(x) = e^xf^1(x) = [e^x] * [x^1]$﻿

﻿$f(x) = In(x) f1 (x) = \frac{1}{x}$﻿ → if ﻿$y = In x , y1 = \frac{1}{x} × \frac{dx}{dx} = \frac{1}{x} × x^1$﻿

﻿$f(x) = b^x f^1(x) = b^x In(b)$﻿

﻿$f(x) = a^x f1(x) = a^x * Ina * x^1$﻿

### Trigonometric Derivative

﻿$\frac{d}{dx} sin x = [cosx] * x^1$﻿

﻿$\frac{d}{dx} cos x = [- sin x] * x^1$﻿

﻿$\frac{d}{dx} tan x = [sec2x] * x^1$﻿

﻿$\frac{d}{dx} csc x = -csc * cot x$﻿

﻿$\frac{d}{dx} sec x = sec * tan x$﻿

﻿$\frac{d}{dx} cot x = -csc2x$﻿

﻿$\frac{d}{dx} arc sin x = \frac{1}{√1-x^2}$﻿

﻿$\frac{d}{dx} arc cos x = \frac{-1}{√ 1-x^2}$﻿

﻿$\frac{d}{dx} arctan x = \frac{1}{1+ x^2}$﻿

﻿$\frac{d}{dx} arccsc x = \frac{-1}{1 * 1 * √ x2 - 1}$﻿

﻿$\frac{d}{dx} arcsec x = \frac{1}{1 * 1 * √x2 - 1}$﻿

﻿$\frac{d}{dx} arccot x = \frac{-1}{1 + x^2}$﻿

### Basic Rule

﻿$\frac{d}{dx} (c) = 0$﻿ , ﻿$\frac{d}{dx} = 1$﻿

﻿$\frac{d}{dx} (f±g) =\frac{df}{dx} ± \frac{dg}{dx}$﻿

﻿$\frac{d}{dx} (fg) = f \frac{dg}{dx} + g \frac{dt}{dx}$﻿