Find dy/dx 4x+5y=xy

Problem

4*x+5*y=x*y

Solution

1. Differentiate both sides of the equation with respect to x treating y as a function of x

2. Apply the sum rule on the left side and the product rule on the right side.

(d(4)*x)/d(x)+(d(5)*y)/d(x)=(d(x)*y)/d(x)

3. Compute the derivatives, using the chain rule for terms involving y

4+5d(y)/d(x)=xd(y)/d(x)+y(1)

4. Group the terms containing d(y)/d(x) on one side of the equation and the remaining terms on the other side.

5d(y)/d(x)−xd(y)/d(x)=y−4

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

d(y)/d(x)*(5−x)=y−4

6. Solve for dy/dx by dividing both sides by (5−x)

d(y)/d(x)=(y−4)/(5−x)

Final Answer

d(y)/d(x)=(y−4)/(5−x)

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