c. solve the equation ( y(x^{2}+4)=8 ) for ( y ) to find an explicit expression for ( y ) and then calculate…

c. solve the equation ( y(x^{2}+4)=8 ) for ( y ) to find an explicit expression for ( y ) and then calculate ( \frac{dy}{dx} ). ( \frac{dy}{dx}=square )

c. solve the equation ( y(x^{2}+4)=8 ) for ( y ) to find an explicit expression for ( y ) and then calculate ( \frac{dy}{dx} ). ( \frac{dy}{dx}=square )

Answer

Explanation:

Step1: Solve for ( y )

Given ( y(x^{2}+4)=8 ), divide both sides by ( x^{2}+4 ) to get ( y=\frac{8}{x^{2}+4}=8(x^{2}+4)^{-1} ).

Step2: Differentiate ( y ) using the chain rule

The chain rule states that if ( y = f(g(x)) ), then ( y^\prime=f^\prime(g(x))\cdot g^\prime(x) ). Let ( u = x^{2}+4 ), so ( y = 8u^{-1} ). First, find ( \frac{dy}{du} ): ( \frac{dy}{du}=-8u^{-2}=-\frac{8}{u^{2}} ). Then, find ( \frac{du}{dx} ): ( \frac{du}{dx}=2x ). By the chain rule ( \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx} ). Substitute ( u = x^{2}+4 ) into the formula: ( \frac{dy}{dx}=-\frac{8}{(x^{2}+4)^{2}}\cdot2x ).

Answer:

( \frac{dy}{dx}=-\frac{16x}{(x^{2}+4)^{2}} )