To determine

To find: The derivative dydx by implicit differentiation.

Answer

The differentiation of xy=x2+y2 is dydx=x−yx2+y2xx2+y2−y.

Explanation

Given:

The equation xy=x2+y2.

Derivative rules:

(1) Chain rule: If y=f(u) and u=g(x)  are both differentiable function, then

dydx=dydu⋅dudx.

(2) Product Rule: If f1(x) and f2(x) are both differentiable, then

ddx[f(x)g(x)]=f(x)ddx(g(x))+g(x)ddxf(x).

Calculation:

Obtain the derivative of xy=x2+y2 implicitly with respect to x.

xy=x2+y2

Differentiate with respect to x on both sides.

ddx(xy)=ddx(x2+y2)

Let u=x2+y2 and apply the chain rule (1).

ddx(xy)=ddx(u)ddx(xy)=ddx(u12)

Apply the product rule (2) and the chain rule (1).

xddx(y)+yddx(x)=ddu(u12)dudxxdydx+y(1)=(12u12−1)dudxxdydx+y=(12u−12)dudx

Substitute the value u=x2+y2 and simplify the expression,

xdydx+y=(12(x2+y2)−12)ddx((x2+y2))xdydx+y=(12(x2+y2)−12)[ddx(x2)+ddx(y2)]xdydx+y=(12(x2+y2)−12)[2x+ddx(y2)]

Apply the chain rule and simplify the terms,

xdydx+y=(12(x2+y2)−12)[2x+(ddy(y2))(dydx)]xdydx+y=x(x2+y2)−12+12(x2+y2)−12(2y)(dydx)xdydx+y=x(x2+y2)−12+y(x2+y2)−12dydxdydx(x−y(x2+y2)−12)=x(x2+y2)−12−y

Multiply the equation by (x2+y2)12.

dydx(x−y(x2+y2)−12)⋅(x2+y2)12=(x(x2+y2)−12−y)⋅(x2+y2)12dydx(x(x2+y2)12−y)=(x−y(x2+y2)12)dydx(xx2+y2−y)=(x−yx2+y2)dydx=x−yx2+y2xx2+y2−y

Therefore, the differentiation of xy=x2+y2 is dydx=x−yx2+y2xx2+y2−y.