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

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Find derivatives of exponential functions

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Q. Find the derivative of

\newline

f(x)

.

\newline

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

Identify Function: Identify the function to differentiate. $\newline$ We are given the function $f(x) = e^{(x+1)}$ , and we need to find its derivative with respect to $x$ .
Apply Chain Rule: Apply the chain rule for differentiation. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In this case, the outer function is $e^u$ (where $u = x+1$ ) and the inner function is $u = x+1$ .
Differentiate Outer Function: Differentiate the outer function with respect to the inner function. $\newline$ The derivative of $e^u$ with respect to $u$ is $e^u$ . So, the derivative of $e^{(x+1)}$ with respect to $x+1$ is $e^{(x+1)}$ .
Differentiate Inner Function: Differentiate the inner function with respect to $x$ . The derivative of $u = x+1$ with respect to $x$ is $1$ , since the derivative of $x$ is $1$ and the derivative of a constant is $0$ .
Multiply Derivatives: Multiply the derivatives from Step $3$ and Step $4$ . $\newline$ According to the chain rule, we multiply the derivative of the outer function by the derivative of the inner function. This gives us the derivative of $f(x)$ with respect to $x$ : $f'(x) = e^{(x+1)} \cdot 1$ .
Simplify Expression: Simplify the expression. $\newline$ Multiplying $e^{(x+1)}$ by $1$ does not change the expression, so the derivative $f'(x)$ remains $e^{(x+1)}$ .

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