Questions: If h(x)=u(x)^v(x), then d/dx h(x)=u(x)^v(x)[v(x) u'(x)/u(x)+(ln(u(x)) v'(x))]. Use the above definition to find the derivative of the following function. h(x)=(x^2+4)^(2x) d/dx h(x)=□

Transcript text: If $h(x)=u(x)^{v(x)}$, then $\frac{d}{d x} h(x)=u(x)^{v(x)}\left[\frac{v(x) u^{\prime}(x)}{u(x)}+\left(\ln (u(x)) v^{\prime}(x)\right)\right]$. Use the above definition to find the derivative of the following function. \[ \begin{array}{l} h(x)=\left(x^{2}+4\right)^{2 x} \\ \frac{d}{d x} h(x)=\square \end{array} \]

Solution

Solution Steps

To find the derivative of the function $ h(x) = (x^2 + 4)^{2x} $ using the given formula, we need to identify $ u(x) $ and $ v(x) $ and then apply the formula. Here, $ u(x) = x^2 + 4 $ and $ v(x) = 2x $. We will then compute $ u'(x) $ and $ v'(x) $, and substitute these into the formula to find the derivative.

Solution Approach

Identify $ u(x) $ and $ v(x) $.
Compute $ u'(x) $ and $ v'(x) $.
Substitute $ u(x) $, $ v(x) $, $ u'(x) $, and $ v'(x) $ into the given derivative formula.
Simplify the expression to get the final derivative.

Step 1: Define the Functions

Let $ u(x) = x^2 + 4 $ and $ v(x) = 2x $.

Step 2: Compute the Derivatives

The derivatives are calculated as follows:

$ u'(x) = \frac{d}{dx}(x^2 + 4) = 2x $
$ v'(x) = \frac{d}{dx}(2x) = 2 $

Step 3: Apply the Derivative Formula

Using the formula for the derivative of $ h(x) = u(x)^{v(x)} $: \[ \frac{d}{dx} h(x) = u(x)^{v(x)} \left[ \frac{v(x) u'(x)}{u(x)} + \ln(u(x)) v'(x) \right] \] Substituting the values: \[ \frac{d}{dx} h(x) = (x^2 + 4)^{2x} \left[ \frac{2x \cdot 2x}{x^2 + 4} + \ln(x^2 + 4) \cdot 2 \right] \]

Step 4: Simplify the Expression

The expression simplifies to: \[ \frac{d}{dx} h(x) = (x^2 + 4)^{2x} \left[ \frac{4x^2}{x^2 + 4} + 2 \ln(x^2 + 4) \right] \] Further simplification yields: \[ \frac{d}{dx} h(x) = 2(x^2 + 4)^{2x - 1} \left[ 2x^2 + (x^2 + 4) \ln(x^2 + 4) \right] \]