Source: algebrica.org - CC BY-NC 4.0 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 of a function at a point using the definition of the difference quotient. We also studied how to differentiate simple functions and composite functions. Now, let’s see how to differentiate power functions of the form:
To calculate the derivative of such a function, a combination of the logarithmic rule and the derivative of exponential functions is used. The general formula for the derivative of $f(x)^g(x)$, with $f$ and $g$ differentiable, is as follows:
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:
For the properties of logarithms $\log_a(b^c) = c \cdot \log_a(b)$
The equality can be rewritten as:
Since $\ln y$ is a composite function, its derivative is
Let’s compute the derivative for the element on the right-hand side of the equality $2x \cdot \ln(x)$:
We obtain:
The equality can be rewritten as:
Since $y = x^{2x}$, we have:
Therefore, the derivative of $y = x^2$ is equal to:
Test yourself
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