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Given $y = e^{kx}$, where $k$ is a constant, find \( \frac{dy}{dx} \) - AQA - A-Level Maths Pure - Question 2 - 2019 - Paper 1
Question 2
Given $y = e^{kx}$, where $k$ is a constant, find \( \frac{dy}{dx} \). Circle your answer.
Worked Solution & Example Answer:Given $y = e^{kx}$, where $k$ is a constant, find \( \frac{dy}{dx} \) - AQA - A-Level Maths Pure - Question 2 - 2019 - Paper 1
Step 1
Find \( \frac{dy}{dx} \)
Answer
To differentiate the function ( y = e^{kx} ), we apply the chain rule of differentiation. The chain rule states that if ( y = e^{u} ) where ( u = kx ), then:
[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} ]
First, we differentiate ( e^u ) with respect to ( u ):
[
\frac{dy}{du} = e^{u} = e^{kx}
]
Next, we differentiate ( u = kx ) with respect to ( x ):
[
\frac{du}{dx} = k
]
Thus, substituting back into the chain rule gives:
[ \frac{dy}{dx} = e^{kx} \cdot k = k e^{kx} ]
Therefore, the correct response is ( \frac{dy}{dx} = k e^{kx} ). This answer should be circled from the options provided.