Find dy/dx ycos(x)=5x^2+2y^2

Problem

y*cos(x)=5*x2+2*y2

Solution

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

(d(y)*cos(x))/d(x)=d(5*x2+2*y2)/d(x)

2. Apply the product rule to the left side, where (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x) and the power rule and chain rule to the right side.

y*(−sin(x))+cos(x)d(y)/d(x)=10*x+4*yd(y)/d(x)

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

cos(x)d(y)/d(x)−4*yd(y)/d(x)=10*x+y*sin(x)

4. Factor out the common factor of d(y)/d(x) from the left side.

d(y)/d(x)*(cos(x)−4*y)=10*x+y*sin(x)

5. Isolate the derivative by dividing both sides by the expression in the parentheses.

d(y)/d(x)=(10*x+y*sin(x))/(cos(x)−4*y)

Final Answer

d(y)/d(x)=(10*x+y*sin(x))/(cos(x)−4*y)

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