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

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Find derivatives using the quotient rule I

Full solution

Q. Find the derivative of

f(x)

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

Find Derivative of $e^{(x+1)}$ : We need to find the derivative of the function $f(x) = e^{(x+1)}$ . The derivative of $e^u$ , where $u$ is a function of $x$ , is $e^u$ times the derivative of $u$ with respect to $x$ . In this case, $u = x + 1$ .
Apply Chain Rule: The derivative of $u = x + 1$ with respect to $x$ is $1$ , since the derivative of a constant is $0$ and the derivative of $x$ is $1$ .
Calculate Final Derivative: Applying the chain rule, the derivative of $f(x)$ with respect to $x$ is $e^{(x+1)}$ times the derivative of $(x+1)$ with respect to $x$ , which we found to be $1$ .
Calculate Final Derivative: Applying the chain rule, the derivative of $f(x)$ with respect to $x$ is $e^{(x+1)}$ times the derivative of $(x+1)$ with respect to $x$ , which we found to be $1$ .Therefore, the derivative of $f(x) = e^{(x+1)}$ is simply $e^{(x+1)} \times 1$ , which simplifies to $e^{(x+1)}$ .

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