Find the Derivative - d/dx f(xy)=75x+80y

Problem

d()/d(x)*(ƒ*(x*y)=75*x+80*y)

Solution

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

2. Apply the chain rule to the left side, where the outer function is ƒ and the inner function is x*y

(d(ƒ)*(x*y))/d(x)=ƒ′*(x*y)⋅(d(x)*y)/d(x)

3. Apply the product rule to the inner derivative (d(x)*y)/d(x)

(d(x)*y)/d(x)=y+xd(y)/d(x)

4. Differentiate the right side of the equation term by term.

d(75*x+80*y)/d(x)=75+80d(y)/d(x)

5. Equate the results from both sides to form the differentiated equation.

ƒ′*(x*y)*(y+xd(y)/d(x))=75+80d(y)/d(x)

6. Distribute the ƒ′*(x*y) term on the left side.

y*ƒ′*(x*y)+x*ƒ′*(x*y)d(y)/d(x)=75+80d(y)/d(x)

7. Isolate the terms containing d(y)/d(x) on one side of the equation.

x*ƒ′*(x*y)d(y)/d(x)−80d(y)/d(x)=75−y*ƒ′*(x*y)

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

d(y)/d(x)*(x*ƒ′*(x*y)−80)=75−y*ƒ′*(x*y)

9. Solve for the derivative by dividing both sides by the coefficient of d(y)/d(x)

d(y)/d(x)=(75−y*ƒ′*(x*y))/(x*ƒ′*(x*y)−80)

Final Answer

d(y)/d(x)=(75−y*ƒ′*(x*y))/(x*ƒ′*(x*y)−80)

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