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

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Find derivatives of exponential functions

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Q. Find the derivative

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, we have $\frac{d}{du}(e^u) = e^u$ .
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 $0$ . So, we have $\frac{du}{dx} = 1$ .
Apply chain rule: Apply the chain rule using the results from steps $3$ and $4$ . $\newline$ The derivative of $f(x)$ with respect to $x$ is the product of the derivatives from steps $3$ and $4$ . Therefore, $f'(x) = e^{(x+1)} \times 1$ .
Simplify derivative: Simplify the expression for the derivative. Since multiplying by $1$ does not change the value, the derivative simplifies to $f'(x) = e^{(x+1)}$ .

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