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

Problem

d(e(−1/x))/d(x)

Solution

1. Identify the outer function as the exponential function eu and the inner function as u=−x(−1)

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

3. Differentiate the inner function −x(−1) using the power rule.

4. Calculate the derivative of the exponent:

(d(−)*x(−1))/d(x)=−(−1)*x(−2)

5. Simplify the resulting expression:

(d(−)*x(−1))/d(x)=1/(x2)

6. Combine the parts to find the final derivative:

d(e(−1/x))/d(x)=e(−1/x)⋅1/(x2)

Final Answer

d(e(−1/x))/d(x)=(e(−1/x))/(x2)

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