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

Problem

d()/d(x)*e^(1/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=1/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 u=x^(−1) using the Power Rule.

d()/d(x)*x^(−1)=−1*x^(−2)

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

d(e^(1/x))/d(x)=e^(1/x)⋅(−x^(−2))

5. Simplify the expression by rewriting the negative exponent as a fraction.

d(e^(1/x))/d(x)=−(e^(1/x))/(x^2)

Final Answer

d(e^(1/x))/d(x)=−(e^(1/x))/(x^2)

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