Suppose that x and y are related by the equation $x ^ { 3 }$ + $( 2 y + 1 ) ^ { 2 }$ = $y ^ { 2 }$ . Use implicit differentiation to determine $\frac { d y } { d x }$ .
A) $\frac { d y } { d x }$ = $\frac { 3 x ^ { 2 } + 4 ( 2 y + 1 ) } { 2 y }$
B) $\frac { d y } { d x }$ = 3 $x ^ { 2 }$ + 2(y + 1)
C) $\frac { d y } { d x }$ = $\frac { - 3 x ^ { 2 } } { 2 ( 3 y + 2 ) }$
D) $\frac { d y } { d x }$ = $\frac { - 3 x ^ { 2 } } { 6 y + 1 }$

Question

Quizplus · Accepted Answer

Differentiating both sides of the equation $x^3 + (2y + 1)^2 = y^2$ with respect to $x$, we get $3x^2 + 2(2y + 1)(2\frac{dy}{dx}) = 2y\frac{dy}{dx}$. Solving for $\frac{dy}{dx}$, we find $\frac{dy}{dx} = \frac{-3x^2}{4(2y + 1) - 2y} = \frac{-3x^2}{8y + 4 - 2y} = \frac{-3x^2}{6y + 4}$, which simplifies to $\frac{-3x^2}{2(3y + 2)}$, matching option C.

Suppose That X and Y Are Related by the Equation x3x ^ { 3 }x3

Suppose That X and Y Are Related by the Equation $x ^ { 3 }$