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Find the derivative of $f(x)$ . $\newline$ $f(x) = e^{(x + 1)}$ $\newline$ $f'\left(x\right) =$ ______

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Find derivatives using the chain rule I

Full solution

Q. Find the derivative of

f(x)

.

\newline

f(x) = e^{(x + 1)}

\newline

f'\left(x\right) =

______

Use Chain Rule: Use the chain rule cuz we got a composite function here. The inside function is $u(x) = x + 1$ and the outside function is $e^u$ .
Derivative of $e^u$ : The derivative of $e^u$ with respect to $u$ is $e^u$ . So, the derivative of the outside function is still $e^{(x + 1)}$ when we plug the inside function back in.
Derivative of $u(x)$ : Now, the derivative of the inside function, $u(x) = x + 1$ , is just $1$ .
Multiply Derivatives: Chain rule says to multiply the derivatives of the outside and inside functions. So, we got $f'(x) = e^{(x + 1)} \times 1$ .
Simplify Result: Simplify that and we end up with $f'(x) = e^{(x + 1)}$ . That's it, we're done!

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