IB Mathematics (SL)/Calculus

Introduction

Average Rate of Change (AROC) between x=a and x=b in f(X)

AROC = $\frac{f(a)-f(b)}{a-b}$

Instantaneous Rate of Change (IROC) at x=a is the slope of the line tangent at x=a:

IROC =$\underset{x\to a}{\lim }\frac{f(x)-f(a)}{x-a}$

Definition of the derivative f'(x) of a function f(x) (First principles of calculus):

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$

Basic Differentiation

The following formulas are shortcuts for finding the derivative

Derivative of a power function

$f(x)=a{x}^n$ , $f^{\prime }(x)=an{x}^{n-1}$

Derivative of exponential function

$f(x)=e^x$ , $f^{\prime }(x)=e^x$

Derivative of logaritmic function

$f(x)=\ln x$ , $f^{\prime }(x)=\frac{1}{x}$

Derivative of trigonometric functions

$f(x)=\sin x$ , $f^{\prime }(x)=\cos x$

$f(x)=\cos x$ , $f^{\prime }(x)=-\sin x$

$f(x)=\tan x$ , $f^{\prime }(x)=\frac{1}{{\cos }^{2}x}={\sec }^{2}x$

Derivatives of sum of two functions

$f(x)=g(x)+h(x)$ , $f^{\prime }(x)=g^{\prime }(x)+h^{\prime }(x)$

Chain Rule

$f(x)=g(u)$ , $f^{\prime }(x)=g^{\prime }(u)\cdot u^{\prime }$

Product Rule

$f(x)=uv$ , $f^{\prime }(x)=u{v}^{\prime }+v{u}^{\prime }$

Quotient Rule

$f(x)=\frac{u}{v}$ , $f^{\prime }(x)=\frac{v{u}^{\prime }-u{v}^{\prime }}{v^2}$