Chain Rule — Question 10

Question 10 - Solution

We are given: $$y = \ln\left( \sqrt{1 + e^{4x}} \right)$$

First, simplify the expression: $$y = \ln\left[(1 + e^{4x})^{1/2}\right] = \frac{1}{2} \ln(1 + e^{4x})$$

Now differentiate: $$\frac{dy}{dx} = \frac{1}{2} \cdot \frac{1}{1 + e^{4x}} \cdot \frac{d}{dx}(1 + e^{4x})$$

Differentiate the inner exponential: $$\frac{d}{dx}(1 + e^{4x}) = 4e^{4x}$$

Putting it all together: $$\frac{dy}{dx} = \frac{1}{2} \cdot \frac{4e^{4x}}{1 + e^{4x}} = \boxed{\frac{2e^{4x}}{1 + e^{4x}}}$$

Nested Function Structure:

$u = e^{4x}$ , $v = 1 + u$ , $w = \ln(\sqrt{v}) = \frac{1}{2} \ln(v)$

Differentiation followed this path: $$\frac{dw}{dv} \cdot \frac{dv}{du} \cdot \frac{du}{dx}$$