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

Problem

d()/d(x)*ecos(x)

Solution

1. Identify the outer and inner functions for the chain rule. The outer function is eu and the inner function is u=cos(x)

2. Apply the chain rule, which states that the derivative of eg(x) is eg(x)⋅d(g(x))/d(x)

3. Differentiate the inner function cos(x) with respect to x which results in −sin(x)

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

5. Simplify the expression by placing the trigonometric term and the negative sign in front.

Final Answer

d(ecos(x))/d(x)=−sin(x)*ecos(x)

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