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

Problem

d(e(−x))/d(x)

Solution

1. Identify the function as a composition of the natural exponential function eu and the linear function u=−x

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

3. Differentiate the inner function u=−x with respect to x which results in −1

4. Multiply the derivative of the outer function by the derivative of the inner function to get e(−x)⋅(−1)

5. Simplify the expression by moving the negative sign to the front.

Final Answer

d(e(−x))/d(x)=−e(−x)

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