Calculus: Differentiation Using First Principles
20 Fully Worked Examples with Step-by-Step Teaching
Introduction
Differentiation is one of the fundamental concepts in calculus. It measures how rapidly a
function changes with respect to its independent variable. Before learning shortcut rules such as
the power rule or product rule, it is essential to understand the first principles (or limit
definition) of differentiation.
The derivative of a function f (x)is defined as
f ( x +h)−f (x )
f ' ( x)=lim
h →0 h
This definition forms the basis of all differentiation techniques.
Example 1
Find the derivative of
f (x)=x+5
using first principles.
Teaching
Step 1: Find f (x+ h).
Replace every x with x +h.
f (x+ h)=(x +h)+5=x+ h+ 5
Step 2: Form the difference.
f ( x+ h)−f (x)¿(x +h+ 5)−( x +5)
Remove the brackets.
¿ x +h+5−x−5¿ h
Step 3: Divide by h .
h
=1
h
Step 4: Evaluate the limit.
lim 1=1
h→ 0
Answer
, '
f ( x)=1
Teaching Note
The derivative of every linear function ax +b is simply its coefficient of x .
Example 2
Find the derivative of
f ( x)=4 x−7
Solution
f (x+ h)=4 (x +h)−7=4 x + 4 h−7
Subtract.
(4 x+ 4 h−7)−(4 x−7)=4 h
Divide by h .
4h
=4
h
Take the limit.
f ' ( x)=4
Teaching Note
Constants disappear because they cancel during subtraction.
Example 3
Find the derivative of
f ( x)=x2
Solution
f ( x+ h)=¿
Expand.
¿ x 2+2 xh+ h2
Subtract.
2 2 2 2
(x + 2 xh+ h )−x =2 xh+ h
Divide by h .
20 Fully Worked Examples with Step-by-Step Teaching
Introduction
Differentiation is one of the fundamental concepts in calculus. It measures how rapidly a
function changes with respect to its independent variable. Before learning shortcut rules such as
the power rule or product rule, it is essential to understand the first principles (or limit
definition) of differentiation.
The derivative of a function f (x)is defined as
f ( x +h)−f (x )
f ' ( x)=lim
h →0 h
This definition forms the basis of all differentiation techniques.
Example 1
Find the derivative of
f (x)=x+5
using first principles.
Teaching
Step 1: Find f (x+ h).
Replace every x with x +h.
f (x+ h)=(x +h)+5=x+ h+ 5
Step 2: Form the difference.
f ( x+ h)−f (x)¿(x +h+ 5)−( x +5)
Remove the brackets.
¿ x +h+5−x−5¿ h
Step 3: Divide by h .
h
=1
h
Step 4: Evaluate the limit.
lim 1=1
h→ 0
Answer
, '
f ( x)=1
Teaching Note
The derivative of every linear function ax +b is simply its coefficient of x .
Example 2
Find the derivative of
f ( x)=4 x−7
Solution
f (x+ h)=4 (x +h)−7=4 x + 4 h−7
Subtract.
(4 x+ 4 h−7)−(4 x−7)=4 h
Divide by h .
4h
=4
h
Take the limit.
f ' ( x)=4
Teaching Note
Constants disappear because they cancel during subtraction.
Example 3
Find the derivative of
f ( x)=x2
Solution
f ( x+ h)=¿
Expand.
¿ x 2+2 xh+ h2
Subtract.
2 2 2 2
(x + 2 xh+ h )−x =2 xh+ h
Divide by h .
