Calculus – Differential & Integral Mathematics Notes
1. Introduction to Calculus
Calculus is the branch of mathematics that studies change and accumulation.
It is divided into:
1. Differential Calculus – Concerned with rates of change and slopes of
curves.
2. Integral Calculus – Deals with accumulation of quantities and areas
under curves.
Applications: Physics (motion, forces), Engineering, Economics, Computer
Science, Statistics.
2. Differential Calculus
2.1 Definition of Derivative
The derivative of a function f(x)f(x)f(x) at a point xxx is:
f′(x)=limh→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}f′
(x)=h→0limhf(x+h)−f(x)
It represents the instantaneous rate of change of the function.
2.2 Basic Derivative Rules
Function Derivative
ccc
(constant) 0
xnx^nxn nxn−1nx^{n-
1}nxn−1
exe^xex exe^xex
lnx\ln xlnx 1/x1/x1/x
sinx\sin xsinx cosx\cos xcosx
cosx\cos
xcosx
−sinx-\sin
x−sinx
Example:
f(x)=3x4+5x2−7⇒f′(x)=12x3+10xf(x) = 3x^4 + 5x^2 - 7 \quad \Rightarrow \
quad f'(x) = 12x^3 + 10xf(x)=3x4+5x2−7⇒f′(x)=12x3+10x
2.3 Rules of Differentiation
