Question: b) \( y = e^{x^3 - 5} \)

Solution

To solve the problem, let’s differentiate the function \( y = e^{x^3 - 5} \). First, apply the chain rule for differentiation. The derivative of \( y = e^{u} \) with respect to \( u \) is \( e^u \). Here, \( u = x^3 - 5 \). Differentiate \( u = x^3 - 5 \) with respect to \( x \): \[ \frac{du}{dx} = 3x^2 \] Apply the chain rule: \[ \frac{dy}{dx} = e^{x^3 - 5} \cdot \frac{du}{dx} \] Substitute \(\frac{du}{dx}\): \[ \frac{dy}{dx} = e^{x^3 - 5} \cdot 3x^2 \] The final answer is: \[ \frac{dy}{dx} = 3x^2 e^{x^3 - 5} \]