Solved Examples on Derivative of Natural log x

Example 1: Find the derivative of the function represented as f(x) = log(x²+4x+5).

Solution:

We have, f(x) = log(x²+4x+5)

By applying chain rule, we get,

⇒ f'(x) = [1/(x²+4x+5)] × d/dx(x²+4x+5)

⇒ f'(x) = [1/(x²+4x+5)] × (2x+4)

⇒ f'(x) = (2x+4)/(x²+4x+5)

Example 2: Find the derivative of the function given by f(x) = 2√(log x).

Solution:

We know that (√x)’ = 1/2√x and (log x)’ = 1/x

For f(x) = 2√ (log x), by applying chain rule, we get,

⇒ f'(x) = 2×[1/2√ (log x)] × d/dx (log x)

⇒ f'(x) = [1/√ (log x)] × (1/x)

⇒ f'(x) = 1/x√ (log x)

Example 3: If a curve is represented as y = log √x, derive an expression for dy/dx.

Solution:

We know that dy/dx is simply the derivative of the function represented by y = f(x).

Therefore, by chain rule,

For y = log √x

⇒ dy/dx = (1/√x) × (1/2√x)

⇒ dy/dx = 1/2x

Thus, for y = log √x, we get dy/dx = 1/2x.

Example 4: Find an expression for slope of the tangent to the curve represented by y = (log x)/x.

Solution:

We know that slope of tangent to the curve is given by the derivative of the function represented as y = f(x). Thus, we need to calculate dy / dx for y = (log x)/x.

By applying quotient rule, we get,

⇒ dy / dx = [x × d/dx (log x) – (log x) × d/dx(x)]/x²

⇒ dy/dx = (1 – log x) / x²

Thus, slope to the tangent of the curve represented by y = (log x)/x is given as (1 – log x)/x².

Example 5: If f(x) = sin (log x), determine the expression for f'(x).

Solution:

We know that (sinx)’ = cosx and (log x)’ = 1/x,

Thus, for f(x) = sin (log x), applying chain rule, we get,

⇒ f'(x) = cos (log x) × (log x)’

⇒ f'(x) = cos (log x)/x

Hence, for f(x) = sin (log x), we have, f'(x) = cos (log x)/x

The derivative of ln x is 1/x. We can also say that the derivative of natural log x is 1/x. The derivative of any function gives the change in the functional value with respect to change in the input variable. Natural log x is an abbreviation for the logarithmic function with the base as Euler’s Number, i.e. e.

In this article, we will discuss the derivative of natural log x, various methods to derive it including the first principal method and implicit differentiation, some solved examples, and practice problems.