lnx^2 derivative

less than a minute read 24-10-2024

Understanding the Derivative of ln(x^2)

The derivative of ln(x^2) might seem a bit daunting at first, but with a little understanding of logarithmic and chain rule properties, it becomes quite straightforward. Let's break down the process step-by-step.

1. Chain Rule: The Key to Differentiating Compositions

The expression ln(x^2) involves a function within a function. We have the natural logarithm function (ln) acting on the function x^2. This calls for the chain rule, which states:

d/dx [f(g(x))] = f'(g(x)) * g'(x)

In our case:

f(x) = ln(x)
g(x) = x^2

2. Finding the Individual Derivatives

Let's find the derivatives of both f(x) and g(x):

f'(x) = 1/x (The derivative of ln(x))
g'(x) = 2x (The derivative of x^2)

3. Applying the Chain Rule

Now, we apply the chain rule to find the derivative of ln(x^2):

d/dx [ln(x^2)] = f'(g(x)) * g'(x)

Substitute f'(x) and g'(x) with their respective values:

= (1/x^2) * 2x

4. Simplifying the Result

Finally, we simplify the expression:

= 2/x

Therefore, the derivative of ln(x^2) is 2/x.

In Summary

The derivative of ln(x^2) is found by applying the chain rule. We first identify the inner and outer functions, find their individual derivatives, and then combine them according to the chain rule. The final result is simplified to obtain the derivative, which is 2/x.