Find dy/dx e^(x/y)=x-y

Problem

e(x/y)=x−y

Solution

1. Differentiate both sides with respect to x using the chain rule and the quotient rule for the exponent.

d(e(x/y))/d(x)=d(x−y)/d(x)

2. Apply the chain rule to the left side and the power rule to the right side.

e(x/y)d()/d(x)x/y=1−d(y)/d(x)

3. Apply the quotient rule to the derivative of x/y

e(x/y)(y(1)−xd(y)/d(x))/(y2)=1−d(y)/d(x)

4. Distribute the term e(x/y) into the fraction.

(y*e(x/y)−x*e(x/y)d(y)/d(x))/(y2)=1−d(y)/d(x)

5. Clear the fraction by multiplying the entire equation by y2

y*e(x/y)−x*e(x/y)d(y)/d(x)=y2−y2d(y)/d(x)

6. Group the terms containing d(y)/d(x) on one side of the equation.

y2d(y)/d(x)−x*e(x/y)d(y)/d(x)=y2−y*e(x/y)

7. Factor out d(y)/d(x) from the left side.

d(y)/d(x)*(y2−x*e(x/y))=y2−y*e(x/y)

8. Solve for the derivative by dividing both sides by the factored term.

d(y)/d(x)=(y2−y*e(x/y))/(y2−x*e(x/y))

Final Answer

d(y)/d(x)=(y2−y*e(x/y))/(y2−x*e(x/y))

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