Calculus Foundations: Part 1
Comprehensive Course on Differential Calculus
Detailed Procedures and Applications
Course Features:
Fundamental Theorems and Definition of Limits
20 Worked Examples with Full Solutions
20 Graded Practice Exercises for Independent Study
Complete Reference Tables for All Standard Functions
1
,1. The Theory of Differentiation
The derivative of a function represents the instantaneous rate at which the function's output
changes relative to its input. In physical terms, it is the velocity of an object moving along a
path defined by the function.
Formal Limit Definition:
f'(x) = limδx→0 [f(x + δx) − f(x)] / δx
Using the alternative notation with h:
f'(x) = limh→0 [f(x + h) − f(x)] / h
Standard Mathematical Notations:
• dy
Leibniz notation:
dx
• Lagrange notation: f'(x) or y'
• Differential operator: Dx[f(x)]
2
,2. Standard Derivative Reference
Function Category f(x) f'(x)
Algebraic xn nxn−1
Constant C 0
ex ex
bx bx ln b
Transcendental
ln x 1/x
loga x 1 / (x ln a)
3
, Standard Derivative Reference (Cont.)
Function Category Function Derivative
sin x cos x
cos x −sin x
tan x sec2 x
Trigonometric
cot x −cosec2 x
sec x sec x tan x
cosec x −cosec x cot x
sinh x cosh x
cosh x sinh x
tanh x sech2 x
Hyperbolic
coth x −cosech2 x
sech x −sech x tanh x
cosech x −cosech x coth x
4
Comprehensive Course on Differential Calculus
Detailed Procedures and Applications
Course Features:
Fundamental Theorems and Definition of Limits
20 Worked Examples with Full Solutions
20 Graded Practice Exercises for Independent Study
Complete Reference Tables for All Standard Functions
1
,1. The Theory of Differentiation
The derivative of a function represents the instantaneous rate at which the function's output
changes relative to its input. In physical terms, it is the velocity of an object moving along a
path defined by the function.
Formal Limit Definition:
f'(x) = limδx→0 [f(x + δx) − f(x)] / δx
Using the alternative notation with h:
f'(x) = limh→0 [f(x + h) − f(x)] / h
Standard Mathematical Notations:
• dy
Leibniz notation:
dx
• Lagrange notation: f'(x) or y'
• Differential operator: Dx[f(x)]
2
,2. Standard Derivative Reference
Function Category f(x) f'(x)
Algebraic xn nxn−1
Constant C 0
ex ex
bx bx ln b
Transcendental
ln x 1/x
loga x 1 / (x ln a)
3
, Standard Derivative Reference (Cont.)
Function Category Function Derivative
sin x cos x
cos x −sin x
tan x sec2 x
Trigonometric
cot x −cosec2 x
sec x sec x tan x
cosec x −cosec x cot x
sinh x cosh x
cosh x sinh x
tanh x sech2 x
Hyperbolic
coth x −cosech2 x
sech x −sech x tanh x
cosech x −cosech x coth x
4
