How to take the Derivative of Ln[f(x)]

The derivative rule for ln[f(x)] is given as:

$$\frac{d}{dx}ln[f(x)] = \frac{f'(x)}{f(x)}$$

Where f(x) is a function of the variable x, and ' denotes the derivative with respect to the variable x.

The derivative rule above is given in terms of a function of x. However, the rule works for single variable functions of y, z, or any other variable. Just replace all instances of x in the derivative rule with the applicable variable. For example, ^d⁄_dθln[f(θ)] = ^f'(θ)⁄_f(θ).

Natural Logarithm Understanding Check

Before we dive deeper into some example problems, let's make sure we have understanding of what the natural logarithm is. The natural logarithm, abbreviated as ln, is a logarithm of base e (Euler’s number). This relation is given as:

lnu = log_eu

The natural logarithm can be written in either form. Ln is the most common way it is written due to being shorter and easier to write.

Derivative of Ln Example Problems

As we can see, taking the derivative of ln requires differentiating the function inside of the natural log and dividing that by the function inside of the natural log. Here are two example problems showing this process in use to take the derivative of ln.

Example Problem 1

Solve ^d⁄_dx[ln(x² + 5)].

Solution:

1. We are taking the natural logarithm of x² + 5, so f(x) = x² + 5. Taking the derivative of that gives us f'(x) = 2x.
2. Now, let's take f(x), f'(x), and plug them into the derivative rule.
   ^d⁄_dxln[f(x)] = ^f'(x)⁄_f(x) = ^2x⁄_{x² + 5}
3. The final answer is ^d⁄_dx ln(x² + 5) = ^2x⁄_{x² + 5}.

Example Problem 2

Solve ^d⁄_dw[log_e(4w)].

Solution:

1. This time the function is of the variable w, so f(w) = 4w and f'(w) = 4.
2. Plugging f(w) and f'(w) into the derivative rule, we get:
   ^d⁄_dw[log_e(4w)] = 4/4w
3. After simplification, the solution is ^d⁄_dw = 1/w.