Given $y = e^{kx}$, where $k$ is a constant, find \( \frac{dy}{dx} \) - AQA - A-Level Maths Mechanics - Question 2 - 2019 - Paper 1

To find the derivative ( \frac{dy}{dx} ) when ( y = e^{kx} ), we apply the chain rule of differentiation:

1. Recognize that the function can be differentiated using the formula for the derivative of the exponential function, which states that if ( y = e^{u} ), then ( \frac{dy}{dx} = e^{u} \cdot \frac{du}{dx} ).

2. Here, let ( u = kx ). Therefore, ( \frac{du}{dx} = k ).

3. Applying the chain rule:

   $\frac{dy}{dx} = e^{kx} \cdot k$

This simplifies to:

$\frac{dy}{dx} = k e^{kx}$

4. Hence, the correct answer is ( \frac{dy}{dx} = k e^{kx} ).