Find dy/dx ycos(x)=4x^2+5y^2

Problem

y*cos(x)=4*x2+5*y2

Solution

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

2. Apply the product rule to the left side, (d(y)*cos(x))/d(x) treating y as a function of x

yd(cos(x))/d(x)+cos(x)d(y)/d(x)=(d(4)*x2)/d(x)+(d(5)*y2)/d(x)

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

−y*sin(x)+cos(x)d(y)/d(x)=8*x+10*yd(y)/d(x)

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

cos(x)d(y)/d(x)−10*yd(y)/d(x)=8*x+y*sin(x)

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

d(y)/d(x)*(cos(x)−10*y)=8*x+y*sin(x)

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

d(y)/d(x)=(8*x+y*sin(x))/(cos(x)−10*y)

Final Answer

d(y)/d(x)=(8*x+y*sin(x))/(cos(x)−10*y)

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