Class notes Differential Calculus

Module 7: LOGARITHMIC AND EXPONENTIAL DIFFERENTIATION

7.1 Derivatives of Logarithmic Functions

If u is a differentiable function for all positive values of x, then
𝑑 1 𝑑𝑢
(log 𝑎 𝑢) =
𝑑𝑥 𝑢𝑙𝑛𝑎 𝑑𝑥

Specifically if a = e, then

𝑑 1 𝑑𝑢
(ln 𝑢) =
𝑑𝑥 𝑢 𝑑𝑥

2𝑥
Example 1 Find the derivative of the function 𝑦 = ln
𝑥+3

Solution:
Applying the formula, we have
𝑑 2𝑥 1 𝑑 2𝑥 𝑥+3 (𝑥+3)2−2𝑥
𝑑𝑥
(ln 𝑥+3) = 2𝑥 ( )=
𝑑𝑥 𝑥+3 2𝑥
[
(𝑥+3)2
]
𝑥+3
𝑑𝑦 6
=
𝑑𝑥 2𝑥(𝑥+3)
𝑑𝑦 3
=
𝑑𝑥 𝑥(𝑥+3)

Alternative solution:
2𝑥
𝑦 = ln = ln(2𝑥) − ln⁡(𝑥 + 3)
𝑥+3
𝑑𝑦 1 1 1 1 𝑥+3−𝑥 3
= (2) − = − = =
𝑑𝑥 2𝑥 𝑥+3 𝑥 𝑥+3 𝑥(𝑥+3) 𝑥(𝑥+3)

−√𝑥 2 +4 1 2+√𝑥 2 +4
Example 2 Determine the derivative of 𝑦 = − ln ( )
2𝑥 2 4 𝑥

Solution:
−𝑥 −2 √𝑥 2 + 4 1
⁡⁡⁡⁡⁡⁡⁡𝑦 = − [ln(2 + √𝑥 2 + 4) − ln 𝑥]
2 4
2𝑥
𝑑𝑦 −1 −2 2𝑥 1 2 1
= [𝑥 ( ) + √𝑥 2 + 4(−2𝑥 −3 )] − [ 2√𝑥 + 4 ] +
𝑑𝑥 2 2√𝑥 2 + 4 4 2 + √𝑥 2 + 4 4𝑥

𝑑𝑦 −1 −3 𝑥 2 − 2(𝑥 2 + 4) 𝑥 1
= 𝑥 ( )− +
𝑑𝑥 2 √𝑥 2 + 4 4√𝑥 2 + 4(2 + √𝑥 2 + 4) 4𝑥
2
𝑑𝑦 𝑥 +8 𝑥 1
=( )− +
𝑑𝑥 2𝑥 3 √𝑥 2 + 4 4√𝑥 2 + 4(2 + √𝑥 2 + 4) 4𝑥

Example 3 Find the derivative of the function 𝑦 = ln √2 + 𝑐𝑜𝑠 2 𝑥

Solution:
1
𝑦 = ln(2 + 𝑐𝑜𝑠 2 𝑥)1/2 = ln(2 + 𝑐𝑜𝑠 2 𝑥)
2
𝑑𝑦 1 2𝑐𝑜𝑠𝑥(−𝑠𝑖𝑛𝑥) 𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑥
= ( 2
)=−
𝑑𝑥 2 2 + 𝑐𝑜𝑠 𝑥 2 + 𝑐𝑜𝑠 2 𝑥

(3𝑥−1)4 (2𝑥+3)3
Example 4 Apply logarithmic differentiation to find the derivative of 𝑦 = (4𝑥+5)5