If y=x( (a+y)) , then find d y d x when x=0

Step-by-step solution

Given the equation y = x (a + y) , we can express x as a function of y : x = y (a + y) Differentiating both sides with respect to y using the quotient rule: dx dy = (a + y) y - y (- (a + y)) ^2(a + y) Using the trigonometric identity A B + A B = (A - B) , we get: dx dy = (a + y - y) ^2(a + y) = a ^2(a + y) Therefore, dy dx = ^2(a + y) a . When x = 0 , substituting into the original equation y = x (a + y) gives y = 0 , which implies y = 0 . Substituting y = 0 into the expression for dy dx : dy dx = ^2(a + 0) a = ^2