Chain Rule — Question 1

Question 1

Differentiate the function: $$f(x) = \sqrt{1 + e^{3x^2}}$$

Clearly show how the chain rule is used in multiple layers.

Question 1 - Solution

We are given: $$f(x) = \sqrt{1 + e^{3x^2}} = (1 + e^{3x^2})^{1/2}$$

Use the chain rule:

$$f'(x) = \frac{1}{2}(1 + e^{3x^2})^{-1/2} \cdot \frac{d}{dx}(1 + e^{3x^2})$$

$$= \frac{1}{2\sqrt{1 + e^{3x^2}}} \cdot \frac{d}{dx}(e^{3x^2})$$

Now apply the chain rule again to $e^{3x^2}$:

$$\frac{d}{dx}(e^{3x^2}) = e^{3x^2} \cdot \frac{d}{dx}(3x^2) = e^{3x^2} \cdot 6x$$

Putting it all together: $$f'(x) = \frac{1}{2\sqrt{1 + e^{3x^2}}} \cdot e^{3x^2} \cdot 6x = \boxed{\frac{3x \cdot e^{3x^2}}{\sqrt{1 + e^{3x^2}}}}$$