Everything Maths Grade 12

CHAPTER 7. DIFFERENTIAL CALCULUS 7.3

Example 7: Derivatives - First Principles

QUESTION

Calculate the derivativeof g(x) = x− 1 from first principles.

SOLUTION

Step 1 : Calculate the gradientat a point

We know that the gradient at a point x is given by:

g�(x) = lim

h→ 0

g(x +h)−g(x)

h

Step 2 : Write g(x +h) and simplify

g(x +h) = x +h− 1

Step 3 : Calculate limit

g�(x) = lim

h→ 0

g(x +h)−g(x)

h

= lim

h→ 0

x +h− 1 − (x− 1)

h

= lim

h→ 0

x +h− 1 −x + 1

h

= lim

h→ 0

h

h

= lim

h→ 0

1

= 1

Step 4 : Write the final answer

The derivative g�(x) of g(x) = x− 1 is 1.

Exercise 7 - 2

1. Given g(x) =−x^2

(a) determine

g(x +h)−g(x)

h

(b) hence, determine

lim

h→ 0

g(x +h)−g(x)

h

(c) explain the meaningof your answer in (b).

2. Find the derivative of f (x) =− 2 x^2 + 3x using first principles.