 Find out the 𝑛𝑡ℎderivative of the following functions:

a) 𝑦 = 𝑒𝑎𝑥

To find the \(n\)th derivative of the function \(y = e^{ax}\), we can proceed as follows:

### Function:
\[ y = e^{ax} \]

### First Derivative (\( y' \)):
\[ y' = \frac{d}{dx}\left(e^{ax}\right) = ae^{ax} \]

### Second Derivative (\( y'' \)):
\[ y'' = \frac{d}{dx}\left(ae^{ax}\right) = a^2e^{ax} \]

### Third Derivative (\( y''' \)):
\[ y''' = \frac{d}{dx}\left(a^2e^{ax}\right) = a^3e^{ax} \]

### General \(n\)th Derivative (\( y^{(n)} \)):
By observing the pattern from the first few derivatives, we can see that the \(n\)th derivative is:
\[ y^{(n)} = a^n e^{ax} \]

So, the \(n\)th derivative of \(y = e^{ax}\) is:
\[ y^{(n)} = a^n e^{ax} \]

Would you like more details on this process or have any further questions?

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Here are 5 related questions that might interest you:

1. What is the \(n\)th derivative of \(y = \sin(ax)\)?
2. How does the \(n\)th derivative of \(y = \cos(ax)\) differ from that of \(y = e^{ax}\)?
3. Can the \(n\)th derivative pattern for \(y = e^{ax}\) be applied to other exponential functions?
4. What is the significance of the constant \(a\) in the function \(y = e^{ax}\) when finding derivatives?
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Learn how to find the nth derivative of the function y = e^{ax} with a step-by-step explanation. This problem covers derivatives of exponential functions and their applications. Suitable for advanced high school students and beyond.

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