Chain Rule — Question 8

Question 8 - Solution

We are given: $$y = \ln\left(\sqrt{1 + e^{3x^2}}\right)$$

We rewrite the square root as a power: $$y = \ln\left( (1 + e^{3x^2})^{1/2} \right)$$

Using the log rule: $$y = \frac{1}{2} \ln(1 + e^{3x^2})$$

Differentiate: $$\frac{dy}{dx} = \frac{1}{2} \cdot \frac{1}{1 + e^{3x^2}} \cdot \frac{d}{dx}[e^{3x^2}]$$

Differentiate the exponential: $$\frac{d}{dx}[e^{3x^2}] = e^{3x^2} \cdot \frac{d}{dx}[3x^2] = e^{3x^2} \cdot 6x$$

Putting it all together: $$\frac{dy}{dx} = \frac{1}{2} \cdot \frac{1}{1 + e^{3x^2}} \cdot e^{3x^2} \cdot 6x$$

Simplify: $$\boxed{ \frac{dy}{dx} = \frac{3x e^{3x^2}}{1 + e^{3x^2}} }$$

Explanation: This function has a nested composition: Innermost: $3x^2$ , Exponential: $e^{3x^2}$ , Added to 1 , Square root → converted to a power , Then natural logarithm

At each step, the derivative is taken from the outside and multiplied by the derivative of the inside, following the chain rule.