Chapter 12: Rules of Differentiation

Question 1

State the simple rules of differentiation

Answer 1

1) Differentiating a constant
- f(x) = c
- f’(x) = 0

2) Differentiating xⁿ
- f(x) = xⁿ
- f’(x) = nxⁿ⁻¹ [power bring down power minus 1]

3) Differentiating the product of a constant and a function
- f(x) = cf(x)
- f’(x) = c x f’(x)

4) Addition/subtraction rule
- f(x) = u(x) + v(x)
- f’(x) = u’(x) + v’(x)

Question 2

State the chain rule

Answer 2

Chain rule:
- y = [f(x)]ⁿ
- dy/dx = n[f(x)]ⁿ⁻¹ x f’(x)

Question 3

State the product rule

Answer 3

Product rule:
- y = u(x)v(x)
- dy/dx = u’(x)v(x)+u(x)v’(x)

Question 4

State the quotient rule

Answer 4

Quotient rule:
- y = u(x)/v(x)
- dy/dx = [u’(x)v(x) - u(x)v’(x)] / [v(x)]²

Question 5

State the rules for the derivatives of exponential functions

Answer 5

Exponential graph (y=aˣ):
- Domain: x∈R
- Range: y>0
- Horizontal asymptote: y=0 (aˣ≠ 0 when a>0)
- y-intercept: (0,1), as a⁰=1

f(x)=eˣ
f’(x)=eˣ

f(x)=eᶠ⁽ˣ⁾
f’(x) = eᶠ⁽ˣ⁾ x f’(x)

Question 6

State the rules for the derivatives of logarithmic functions

Answer 6

Logarithmic rules:
1) ln(ab) = ln(a)+ln(b)
2) ln(a/b) = ln(a)-ln(b)
3) ln(aⁿ)=n ln(a)
4) lne=1
5) ln1=0

Logarithmic function (y=logx)
- Inverse function of y=aˣ
- Domain: x>0
- Range: y∈R
- Vertical asymptote: x=0
- x-intercept: (1,0), as log1=0

f(x) = ln(x)
f’(x) = 1/x

f(x) = ln f(x)
f’(x)
= (1/f(x) ) x f’(x)
= f’(x) / f(x)

Question 7

State the rules for the derivatives of trigonometric functions

Answer 7

• f(x) = sin(x)
• f’(x) = cos(x)
• f(x) = sin f(x)
• f’(x) = cos f(x) x f’(x)
• f(x) = cos(x)
• f’(x) = -sin(x)
• f(x) = cos f(x)
• f’(x) = -sin(x) x f’(x)

Question 8

State and prove that:
d/dx (tan x) = (sec x )²

Answer 8

y = tan(x)
y = sin(x) / cos(x)

dy/dx
= [cos(x) x cos(x) - sin(x) x -sin(x)] / [cos(x)]²
= [cos²x + sin²x]/cos²x
= 1/cos²x
= (1/cosx)²
= (sec x )²
= sec²x

Question 9

Explain and find second derivatives

Answer 9

f’(x) OR dy/dx: gradient function
f”(x) OR d²y/dx²: rate of change of f’(x)