To determine
To find: The derivative dydx by implicit differentiation.
Answer
The differentiation of the equation is dydx=x−2y2x−y.
Explanation
Given:
The curve x2−4xy+y2=4.
Derivative rules:
(1) Product Rule: ddx[f1(x)f2(x)]=f1(x)ddx[f2(x)]+f2(x)ddx[f1(x)].
(2) Chain rule: If y=f(u) and u=g(x) are both differentiable function, then
dydx=dydu⋅dudx.
Calculation:
Obtain the differentiation of x2−4xy+y2=4 implicitly with respect to x,
x2−4xy+y2=4
Differentiate with respect to x on both sides,
ddx(x2−4xy+y2)=ddx(4)ddx(x2)−4ddx(xy)+ddx(y2)=ddx(4)
Apply the product rule (1) and the chain rule (2),
ddx(x2)−4ddx(xy)+ddx(y2)=ddx(4)2x−4[xddx(y)+yddx(x)]+[ddy(y2)dydx]=ddx(4)2x−4[xdydx+y(1)]+2ydydx=02x−4xdydx−4y+2ydydx=0
Group the corresponding term simplify further,
2x−4y+dydx(2y−4x)=0dydx(2y−4x)=−2x+4y
Divide the equation by 2y−4x,
dydx=−2x+4y−4x+2y=−2(x−2y)−2(2x−y)=x−2y2x−y
Therefore, the derivative of the equation is dydx=x−2y2x−y.