Answer:
[tex] \displaystyle d)\frac{d ^{2} y}{d{x}^{2} } = 2 + \frac{ 42}{ {x}^{4} }[/tex]
Step-by-step explanation:
we would like to figure out the second derivative of the following:
[tex] \displaystyle y = \frac{ {x}^{4} + 7}{ {x}^{2} } [/tex]
we can rewrite it thus rewrite:
[tex] \displaystyle y = {x}^{2} + 7 {x}^{ - 2} [/tex]
take derivative in both sides:
[tex] \displaystyle \frac{dy}{dx} = \frac{d}{dx}( {x}^{2} + 7 {x}^{ - 2} )[/tex]
by sum derivation we obtain:
[tex] \displaystyle \frac{dy}{dx} = \frac{d}{dx}{x}^{2} + \frac{d}{dx} 7 {x}^{ - 2}[/tex]
by exponent derivation we acquire:
[tex] \displaystyle \frac{dy}{dx} = 2{x}^{} - 14 {x}^{ - 3}[/tex]
take derivative In both sides once again:
[tex] \displaystyle \frac{d ^{2} y}{d{x}^{2} } = \frac{d}{d x } (2{x}^{} - 14 {x}^{ - 3})[/tex]
use difference rule which yields:
[tex] \displaystyle \frac{d ^{2} y}{d{x}^{2} } = \frac{d}{d x } 2{x}^{} - \frac{d}{dx} 14{x}^{ - 3}[/tex]
use exponent derivation which yields:
[tex] \displaystyle \frac{d ^{2} y}{d{x}^{2} } = 2 + 42{x}^{ - 4}[/tex]
by law of exponent we get:
[tex] \displaystyle \frac{d ^{2} y}{d{x}^{2} } = 2 + \frac{ 42}{ {x}^{4} }[/tex]
hence, our answer is d)
