Find dy/dx square root of x+y=1+x^2y^2

Problem

√(,x+y)=1+x2*y2

Solution

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

d(√(,x+y))/d(x)=d(1+x2*y2)/d(x)

2. Apply the chain rule to the left side and the product rule to the right side.

1/(2√(,x+y))⋅(1+d(y)/d(x))=0+2*x*y2+x2⋅2*yd(y)/d(x)

3. Distribute the term on the left side to isolate the derivative terms.

1/(2√(,x+y))+1/(2√(,x+y))d(y)/d(x)=2*x*y2+2*x2*yd(y)/d(x)

4. Group all terms containing d(y)/d(x) on one side and the remaining terms on the other side.

1/(2√(,x+y))d(y)/d(x)−2*x2*yd(y)/d(x)=2*x*y2−1/(2√(,x+y))

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

d(y)/d(x)*(1/(2√(,x+y))−2*x2*y)=2*x*y2−1/(2√(,x+y))

6. Solve for the derivative by dividing both sides by the expression in the parentheses.

d(y)/d(x)=(2*x*y2−1/(2√(,x+y)))/(1/(2√(,x+y))−2*x2*y)

7. Simplify the fraction by multiplying the numerator and denominator by 2√(,x+y)

d(y)/d(x)=(4*x*y2√(,x+y)−1)/(1−4*x2*y√(,x+y))

Final Answer

d(y)/d(x)=(4*x*y2√(,x+y)−1)/(1−4*x2*y√(,x+y))

Want more problems? Check here!