Find the Derivative - d/dy e^(x/y)

Problem

d(e(x/y))/d(y)

Solution

1. Identify the rule needed for the derivative, which is the chain rule for the exponential function eu where u=x/y

2. Apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function with respect to y

d(e(x/y))/d(y)=e(x/y)⋅d()/d(y)x/y

3. Differentiate the inner expression x/y with respect to y treating x as a constant.

d()/d(y)x/y=x⋅d()/d(y)*y(−1)

4. Apply the power rule to find the derivative of y(−1)

x⋅(−1)*y(−2)=−x/(y2)

5. Combine the results to find the final derivative.

d(e(x/y))/d(y)=e(x/y)⋅(−x/(y2))

Final Answer

d(e(x/y))/d(y)=−(x*e(x/y))/(y2)

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