Derivative of Function to Power Function Form

When dealing with derivatives in calculus, one often encounters functions raised to a power. Understanding how to differentiate these functions is crucial for solving a wide range of problems. In this guide, we will explore the differentiation of functions in the form of $f(x)^{g(x)}$, where both $f(x)$ and $g(x)$ are differentiable functions of $x$.

Basic Power Rule

Before diving into the more complex cases, let's recall the basic power rule for differentiation. If $f(x) = x^n$ where $n$ is a constant, then the derivative of $f(x)$ with respect to $x$ is given by:

$$ f'(x) = \frac{d}{dx}x^n = nx^{n-1} $$

Derivative of a Function Raised to a Power

When a function $u(x)$ is raised to the power of a constant $n$, the power rule can be extended as follows:

$$ \frac{d}{dx}u(x)^n = nu(x)^{n-1} \cdot u'(x) $$

This is a direct application of the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

Derivative of a Power Function Form $f(x)^{g(x)}$

When we have a function raised to another function, $f(x)^{g(x)}$, the differentiation becomes more complex. To differentiate this form, we use logarithmic differentiation. Here are the steps:

1. Take the natural logarithm of both sides: $$ \ln y = \ln \left[f(x)^{g(x)}\right] $$

2. Simplify using logarithm properties: $$ \ln y = g(x) \cdot \ln f(x) $$

3. Differentiate both sides with respect to $x$, applying the product rule to the right-hand side: $$ \frac{1}{y} \cdot \frac{dy}{dx} = g'(x) \cdot \ln f(x) + g(x) \cdot \frac{1}{f(x)} \cdot f'(x) $$

4. Solve for $\frac{dy}{dx}$: $$ \frac{dy}{dx} = y \cdot \left[g'(x) \cdot \ln f(x) + g(x) \cdot \frac{f'(x)}{f(x)}\right] $$

5. Substitute back $y = f(x)^{g(x)}$ to get the derivative in terms of $x$: $$ \frac{d}{dx}f(x)^{g(x)} = f(x)^{g(x)} \cdot \left[g'(x) \cdot \ln f(x) + g(x) \cdot \frac{f'(x)}{f(x)}\right] $$

Table of Differences and Important Points

Property	Basic Power Rule $x^n$	Function to a Constant Power $u(x)^n$	Function to a Function Power $f(x)^{g(x)}$
Rule	$nx^{n-1}$	$nu(x)^{n-1} \cdot u'(x)$	$f(x)^{g(x)} \cdot \left[g'(x) \cdot \ln f(x) + g(x) \cdot \frac{f'(x)}{f(x)}\right]$
Requires Chain Rule	No	Yes	Yes
Requires Logarithmic Differentiation	No	No	Yes
Example	$\frac{d}{dx}x^3 = 3x^2$	$\frac{d}{dx}(2x+1)^3 = 3(2x+1)^2 \cdot 2$	$\frac{d}{dx}(x^2)^{\sin x} = (x^2)^{\sin x} \cdot \left[\cos x \cdot \ln x^2 + \sin x \cdot \frac{2x}{x^2}\right]$

Examples

Example 1: Function to a Constant Power

Differentiate $h(x) = (3x^2 + 2x + 1)^5$.

Using the extended power rule:

$$ h'(x) = 5(3x^2 + 2x + 1)^4 \cdot (6x + 2) $$

Example 2: Function to a Function Power

Differentiate $k(x) = (x + 1)^{x}$.

Using logarithmic differentiation:

1. $\ln k = x \ln(x + 1)$
2. $\frac{1}{k} \cdot k' = \ln(x + 1) + \frac{x}{x + 1}$
3. $k' = k \left[\ln(x + 1) + \frac{x}{x + 1}\right]$
4. $k' = (x + 1)^{x} \left[\ln(x + 1) + \frac{x}{x + 1}\right]$

By understanding these rules and methods, students can tackle a wide variety of differentiation problems involving power functions on their exams.