# Derivative of Composite Power Functions

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https://algebrica.org/derivative-of-composite-power-functions/
Fetched from algebrica.org post 6176; source modified 2025-10-29T19:09:06.

## Composite Power Functions and Derivatives

We have previously introduced how to calculate the [derivative](../derivatives.md) of a function at a point using the definition of the [difference quotient](../difference-quotient.md). We also studied how to differentiate simple functions and [composite functions](../the-derivative-of-a-composite-function.md). Now, let’s see how to differentiate power functions of the form:

\\[
D\[f(x)\]^{g(x)}
\\]

To calculate the derivative of such a function, a combination of the [logarithmic](../logarithms.md) rule and the derivative of [exponential functions](../exponential-function.md) is used. The general formula for the derivative of \\(f(x)^g(x)\\), with \\(f\\) and \\(g\\) differentiable, is as follows:

\\[
D\[f(x)\]^{g(x)} = f(x)^{g(x)} \\left\[ g’(x) \\ln f(x) + g(x) \\frac{f’(x)}{f(x)} \\right \]
\\]

Where:

-   \\( f(x)^{g(x)} \\) is the original function.
-   \\( f’(x) \\) is the derivative of \\( f(x) \\).
-   \\( \\ln f(x) \\) is the natural logarithm of \\( f(x) \\).
-   \\( g’(x) \\) is the derivative of \\( g(x) \\).

## Example

Let’s consider the function \\( y = x^{2x} \\) as an example, and calculate its derivative.

* * *

First, let’s rewrite the function by applying the logarithm to both sides:

\\[
\\ln y = \\ln(x^{2x})
\\]

For the properties of logarithms \\(\\log\_a(b^c) = c \\cdot \\log\_a(b)\\)

The equality can be rewritten as:

\\[
\\ln y = 2x \\cdot \\ln(x)
\\]

* * *

Since \\(\\ln y\\) is a composite function, its derivative is

\\[
\\frac{1}{y} \\cdot y’
\\]

Let’s compute the derivative for the element on the right-hand side of the equality \\(2x \\cdot \\ln(x)\\):

\\[
2 \\cdot \\ln(x) + 2x \\cdot \\frac{1}{x}
\\]

We obtain:

\\[
\\frac{1}{y} \\cdot y’ = 2 \\cdot \\ln(x) + 2x \\cdot \\frac{1}{x}
\\]

* * *

The equality can be rewritten as:

\\[
y’ = y \\cdot (2 \\cdot \\ln(x) + 2)
\\]

Since \\(y = x^{2x}\\), we have:

\\[
y’ = x^{2x} \\cdot (2 \\cdot \\ln(x) + 2)
\\]

Therefore, the derivative of \\( y = x^2 \\) is equal to:

\\[
x^{2x} \\cdot (2 \\cdot \\ln(x) + 2)
\\]

## Test yourself

-

\\[
\\text{1. } \\quad y = x^{2\\cos(x)}
\\]

 [solution](../derivative-a-1.md)

-

\\[
\\text{2. } \\quad y = x^{\\ln(x)}
\\]

 [solution](../derivative-a-2.md)


##### The proposed functions are designed to help you consolidate your understanding of composite function derivatives. Try solving them independently before checking the solutions provided.