Examples of the derivatives of logarithmic functions, in calculus, are presented. Several examples, with detailed solutions, involving products, sums and quotients of exponential functions are examined.
The derivative of f(x) = log_{ b} x is given by
f '(x) = 1 / (x ln b)
Note: if f(x) = ln x , then f '(x) = 1 / x
Example 1: Find the derivative of f(x) = log_{ 3} x
Solution to Example 1:

Apply the formula above to obtain
f '(x) = 1 / (x ln 3)
Example 2: Find the derivative of f(x) = ln x + 6x^{ 2}
Solution to Example 2:

Let g(x) = ln x and h(x) = 6x^{ 2}, function f is the sum of functions g and h: f(x) = g(x) + h(x). Use the sum rule, f '(x) = g '(x) + h '(x), to find the derivative of function f
f '(x) = 1 / x + 12x
Example 3: Find the derivative of f(x) = log_{ 3} x / ( 1  x )
Solution to Example 3:

Let g(x) = log_{ 3} x and h(x) = 1  x, function f is the quotient of functions g and h: f(x) = g(x) / h(x). Hence we use the quotient rule, f '(x) = [ h(x) g '(x)  g(x) h '(x) ] / h(x)^{ 2}, to find the derivative of function f.
g '(x) = 1 / (x ln 3)
h '(x) = 1
f '(x) = [ h(x) g '(x)  g(x) h '(x) ] / h(x)^{ 2}
= [ (1  x)(1 / (x ln 3))  (log_{ 3} x)(1) ] / (1  x)^{ 2}
Example 4: Find the derivative of f(x) = ln (4x + 1)
Solution to Example 4:

Let u = 4x + 1 and y = ln u, Use the chain rule to find the derivative of function f as follows.
f '(x) = (dy / du) (du / dx)

dy / du = 1 / u and du / dx = 4
f '(x) = (1 / u)(4) = 4 / u

Substitute u = 4x + 1 in f '(x) above
f '(x) = 4 / (4x + 1)
Exercises Find the derivative of each function.
1  f(x) = ln(x^{ 2})
2  g(x) = ln x  x^{ 7}
3  h(x) = ln x / (2x  3)
4  j(x) = ln (x + 3) ln (x  1)
solutions to the above exercises
1  f '(x) = 2 / x
2  g '(x) = 1 / x 7x^{ 6}
3  h '(x) = (2x  3  2x ln x) / [ x(2x 3)^{ 2} ]
4  j '(x) = ln (x + 3) / (x  1) + ln (x  1) / (x + 3)
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Updated: February 2015
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