Find dy/dx xe^y-10x+3y=0

Problem

x*ey−10*x+3*y=0

Solution

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

d()/d(x)*(x*ey−10*x+3*y)=d()/d(x)*0

2. Apply the product rule to the first term x*ey and the sum rule to the remaining terms.

(d(x)*ey)/d(x)−(d(10)*x)/d(x)+(d(3)*y)/d(x)=0

3. Compute the derivatives using the product rule (u*v)′=u′*v+u*v′ for x*ey and the chain rule for terms involving y

ey+x*eyd(y)/d(x)−10+3d(y)/d(x)=0

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

x*eyd(y)/d(x)+3d(y)/d(x)=10−ey

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

d(y)/d(x)*(x*ey+3)=10−ey

6. Solve for dy/dx by dividing both sides by the expression in the parentheses.

d(y)/d(x)=(10−ey)/(x*ey+3)

Final Answer

d(y)/d(x)=(10−ey)/(x*ey+3)

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