Chain Rule — Question 9

Question 9 - Solution

We are given: $$f(x) = \sin^2(5x^3 + 1)$$

This can be rewritten as: $$f(x) = [\sin(5x^3 + 1)]^2$$

Layers of function composition:

1. Innermost function: $u(x) = 5x^3 + 1$ 2. Middle function: $v(u) = \sin(u)$ 3. Outer function: $f(v) = v^2$

Differentiate step by step:

Start from the outside: $$f'(x) = 2 \cdot \sin(5x^3 + 1) \cdot \frac{d}{dx}[\sin(5x^3 + 1)]$$

Now differentiate the inner sine function: $$\frac{d}{dx}[\sin(5x^3 + 1)] = \cos(5x^3 + 1) \cdot \frac{d}{dx}[5x^3 + 1] = \cos(5x^3 + 1) \cdot 15x^2$$

Putting it all together: $$f'(x) = 2 \cdot \sin(5x^3 + 1) \cdot \cos(5x^3 + 1) \cdot 15x^2$$

$$\boxed{ f'(x) = 30x^2 \cdot \sin(5x^3 + 1) \cdot \cos(5x^3 + 1) }$$

Alternatively, you may express the result using the identity: $$2 \sin\theta \cos\theta = \sin(2\theta)$$

$$\Rightarrow f'(x) = 15x^2 \cdot \sin(2(5x^3 + 1)) = \boxed{15x^2 \cdot \sin(10x^3 + 2)}$$

Either boxed expression is valid.