Find dy/dx e^(x/y)=3x-y

Problem

e(x/y)=3*x−y

Solution

1. Differentiate both sides with respect to x using implicit differentiation.

d(e(x/y))/d(x)=d(3*x−y)/d(x)

2. Apply the chain rule to the left side and differentiate the right side term by term.

e(x/y)⋅d()/d(x)x/y=3−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)=3−d(y)/d(x)

4. Distribute the terms on the left side to isolate the derivative.

(y*e(x/y))/(y2)−(x*e(x/y)d(y)/d(x))/(y2)=3−d(y)/d(x)

5. Simplify the first fraction and move all terms containing d(y)/d(x) to one side.

(e(x/y))/y−3=(x*e(x/y))/(y2)d(y)/d(x)−d(y)/d(x)

6. Factor out d(y)/d(x) from the right side.

(e(x/y)−3*y)/y=d(y)/d(x)*((x*e(x/y))/(y2)−1)

7. Find a common denominator for the expression inside the parentheses.

(e(x/y)−3*y)/y=d(y)/d(x)*((x*e(x/y)−y2)/(y2))

8. Solve for dy/dx by multiplying both sides by the reciprocal of the coefficient.

d(y)/d(x)=(e(x/y)−3*y)/y⋅(y2)/(x*e(x/y)−y2)

9. Simplify the expression by canceling y

d(y)/d(x)=(y*(e(x/y)−3*y))/(x*e(x/y)−y2)

Final Answer

d(y)/d(x)=(y*e(x/y)−3*y2)/(x*e(x/y)−y2)

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