Chain Rule — Question 3

Question 3

Let $y = \ln\left( \tan^3(5x^2 + 1) \right)$.

• Differentiate $y$ with respect to $x$ using the chain rule.

• Simplify the derivative as much as possible.

Question 3 - Solution

We are given: $$y = \ln\left( \tan^3(5x^2 + 1) \right)$$

Step 1: Use logarithmic identity:

$$\ln\left( \tan^3(5x^2 + 1) \right) = 3 \ln\left( \tan(5x^2 + 1) \right)$$

Step 2: Differentiate using chain rule:

$$\frac{dy}{dx} = 3 \cdot \frac{1}{\tan(5x^2 + 1)} \cdot \sec^2(5x^2 + 1) \cdot (10x)$$

Step 3: Simplify the expression:

$$\frac{dy}{dx} = \frac{30x \sec^2(5x^2 + 1)}{\tan(5x^2 + 1)}$$

$$\boxed{ \frac{dy}{dx} = \frac{30x \sec^2(5x^2 + 1)}{\tan(5x^2 + 1)} }$$