Find the Derivative - d/dx 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 e^u and the inner function is u=1/x

2. Apply the Chain Rule which states that the derivative of e^ƒ(x) is e^ƒ(x)⋅d(ƒ(x))/d(x)

d(e^(1/x))/d(x)=e^(1/x)⋅d()/d(x)1/x

3. Rewrite the inner function using a negative exponent to make differentiation easier.

1/x=x^(−1)

4. Apply the Power Rule to find the derivative of x^(−1) which results in −1*x^(−2)

d(x^(−1))/d(x)=−x^(−2)

5. Simplify the expression by converting the negative exponent back into a fraction.

−x^(−2)=−1/(x^2)

6. Combine the results to find the final derivative.

d(e^(1/x))/d(x)=e^(1/x)⋅(−1/(x^2))

Final Answer

d(e^(1/x))/d(x)=−(e^(1/x))/(x^2)

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