Chain Rule — Question 5

Question 5

Let $f(x) = e^{\tan(x^3 + 1)}$.

• Use the chain rule to find $f'(x)$.

• Clearly identify each function used in the composition and the order in which the chain rule is applied.

Question 5 - Solution

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

This is a composition of multiple functions. Let’s break it down:

• Outer function: $u = e^v$

• Middle function: $v = \tan(w)$

• Inner function: $w = x^3 + 1$

Using the chain rule: $$f'(x) = \frac{d}{dx} \left[ e^{\tan(x^3 + 1)} \right] = e^{\tan(x^3 + 1)} \cdot \frac{d}{dx} \left[ \tan(x^3 + 1) \right]$$

Now, $$\frac{d}{dx} \left[ \tan(x^3 + 1) \right] = \sec^2(x^3 + 1) \cdot \frac{d}{dx}[x^3 + 1] = \sec^2(x^3 + 1) \cdot 3x^2$$

Putting it all together: $$f'(x) = e^{\tan(x^3 + 1)} \cdot \sec^2(x^3 + 1) \cdot 3x^2$$

$$\boxed{ f'(x) = 3x^2 \cdot \sec^2(x^3 + 1) \cdot e^{\tan(x^3 + 1)} }$$