Find the Derivative - d/dx f(x)=e^(1/x)

Problem

d(e(1/x))/d(x)

Solution

1. Identify the outer function and the inner function to apply the Chain Rule. The outer function is eu and the inner function is u=1/x

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

3. Differentiate the inner function u=x(−1) using the Power Rule.

d()/d(x)1/x=−x(−2)

4. Substitute the derivative of the inner function back into the Chain Rule expression.

d(e(1/x))/d(x)=e(1/x)⋅−1/(x2)

5. Simplify the expression by combining the terms.

d(e(1/x))/d(x)=−(e(1/x))/(x2)

Final Answer

d(e(1/x))/d(x)=−(e(1/x))/(x2)

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