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

Problem

x*e^y−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*e^y−10*x+3*y)=d()/d(x)*0

2. Apply the product rule to the first term x*e^y and the sum rule to the remaining terms.

(d(x)*e^y)/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*e^y and the chain rule for terms involving y

e^y+x*e^yd(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*e^yd(y)/d(x)+3d(y)/d(x)=10−e^y

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

d(y)/d(x)*(x*e^y+3)=10−e^y

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

d(y)/d(x)=(10−e^y)/(x*e^y+3)

Final Answer

d(y)/d(x)=(10−e^y)/(x*e^y+3)

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