Question: b) \( y = e^{x^3 - 5} \) Problem #2 (5 pts each)

Solution

To solve the problem \( y = e^{x^3} - 5 \), differentiate it with respect to \( x \). The original function is \[ y = e^{x^3} - 5 \] Differentiate using the chain rule for the term \( e^{x^3} \). The derivative of \( e^{u} \) with respect to \( u \) is \( e^{u} \), where \( u = x^3 \). Differentiate \( u \) with respect to \( x \). \[ \frac{du}{dx} = 3x^2 \] Now apply the chain rule: \[ \frac{dy}{dx} = \frac{d}{dx}(e^{x^3}) = e^{x^3} \cdot \frac{du}{dx} = e^{x^3} \cdot 3x^2 \] The derivative of the constant \(-5\) is 0. Thus, the derivative is \[ \frac{dy}{dx} = 3x^2 e^{x^3} \]