If $( x + y ) ^ { 2 } = ( 2 x + 3 ) ^ { 3 }$ , use implicit differentiation to find $\frac { d y } { d x }$ .
A) $\frac { 3 ( 2 x + 3 ) ^ { 2 } } { x + y }$
B) $\frac { 3 ( 2 x + 3 ) ^ { 2 } - x - y } { x + y }$
C) $\frac { 1 - 2 x - 2 y } { 2 x + 2 y }$
D) $\frac { 6 ( 2 x + 3 ) ^ { 2 } } { ( x + y ) ^ { 2 } }$

Question

Quizplus · Accepted Answer

To find $\frac{dy}{dx}$, we differentiate both sides of the equation $(x + y)^2 = (2x + 3)^3$ with respect to $x$, using the chain rule and the product rule as necessary. Differentiating the left side gives $2(x + y)(1 + \frac{dy}{dx})$, and differentiating the right side gives $3(2x + 3)^2(2)$. Setting these equal and solving for $\frac{dy}{dx}$ gives $\frac{dy}{dx} = \frac{3(2x + 3)^2 - (x + y)}{x + y}$.

If (x+y)2=(2x+3)3( x + y ) ^ { 2 } = ( 2 x + 3 ) ^ { 3 }(x+y)2=(2x+3)3

If $( x + y ) ^ { 2 } = ( 2 x + 3 ) ^ { 3 }$