Find the Derivative - d/dx f(x)=e^(sin(x))

Problem

d(e^sin(x))/d(x)

Solution

1. Identify the outer function and the inner function to apply the Chain Rule. The outer function is e^u and the inner function is u=sin(x)

2. Apply the Chain Rule which states that d(e^u)/d(x)=e^u⋅d(u)/d(x)

3. Differentiate the inner function sin(x) with respect to x

d(sin(x))/d(x)=cos(x)

4. Multiply the derivative of the outer function by the derivative of the inner function.

e^sin(x)⋅cos(x)

Final Answer

d(e^sin(x))/d(x)=e^sin(x)*cos(x)

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