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

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Full solution

Q. Find the derivative f(x)=e(x+1) f(x)=e^{(x+1)}
  1. Identify Function: Identify the function to differentiate. The function is f(x)=e(x+1)f(x)=e^{(x+1)}.
  2. Recognize Exponential Function: Recognize that the function is an exponential function with a base of ee and an inner function of (x+1)(x+1).
  3. Apply Chain Rule: Apply the chain rule for differentiation. The chain rule states that the derivative of eue^u, where uu is a function of xx, is eue^u times the derivative of uu with respect to xx. In this case, u(x)=x+1u(x) = x+1 and the derivative of uu with respect to xx is 11.
  4. Differentiate Function: Differentiate the function. The derivative of f(x)f(x) with respect to xx is f′(x)=e(x+1)⋅1f'(x) = e^{(x+1)} \cdot 1.
  5. Simplify Derivative: Simplify the derivative expression. The derivative of f(x)=e(x+1)f(x) = e^{(x+1)} is f′(x)=e(x+1)f′(x) = e^{(x+1)}.

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