Chain Rule — Question 7

Question 7 - Solution

We are given: $$y = e^{\tan^3(2x)} = e^{(\tan(2x))^3}$$

Let’s identify the layers of composition:

• Outermost: $e^u$

• Middle: $u = v^3$

• Inner: $v = \tan(2x)$

• Innermost: $2x$

Apply the chain rule step-by-step:

$$\frac{dy}{dx} = \frac{d}{dx} \left( e^{(\tan(2x))^3} \right) = e^{(\tan(2x))^3} \cdot \frac{d}{dx} \left( (\tan(2x))^3 \right)$$

Differentiate the power: $$\frac{d}{dx} \left( (\tan(2x))^3 \right) = 3(\tan(2x))^2 \cdot \frac{d}{dx}[\tan(2x)]$$

Differentiate $\tan(2x)$: $$\frac{d}{dx}[\tan(2x)] = \sec^2(2x) \cdot 2$$

Now put it all together: $$\frac{dy}{dx} = e^{(\tan(2x))^3} \cdot 3(\tan(2x))^2 \cdot \sec^2(2x) \cdot 2$$

Simplify: $$\boxed{ \frac{dy}{dx} = 6(\tan(2x))^2 \sec^2(2x) \cdot e^{(\tan(2x))^3} }$$