Mathematics: Find the second derivative of the function.

I really dont knowStep-by-step explanation:...

Find the second derivative of the function. - id13680150, helpmewithmath70

rockstargirl9869

09.07.2021

$\displaystyle d)\frac{d ^{2} y}{d{x}^{2} } = 2 + \frac{ 42}{ {x}^{4} }$

Step-by-step explanation:

we would like to figure out the second derivative of the following:

$\displaystyle y = \frac{ {x}^{4} + 7}{ {x}^{2} }$

we can rewrite it thus rewrite:

$\displaystyle y = {x}^{2} + 7 {x}^{ - 2}$

take derivative in both sides:

$\displaystyle \frac{dy}{dx} = \frac{d}{dx}( {x}^{2} + 7 {x}^{ - 2} )$

by sum derivation we obtain:

$\displaystyle \frac{dy}{dx} = \frac{d}{dx}{x}^{2} + \frac{d}{dx} 7 {x}^{ - 2}$

by exponent derivation we acquire:

$\displaystyle \frac{dy}{dx} = 2{x}^{} - 14 {x}^{ - 3}$

take derivative In both sides once again:

$\displaystyle \frac{d ^{2} y}{d{x}^{2} } = \frac{d}{d x } (2{x}^{} - 14 {x}^{ - 3})$

use difference rule which yields:

$\displaystyle \frac{d ^{2} y}{d{x}^{2} } = \frac{d}{d x } 2{x}^{} - \frac{d}{dx} 14{x}^{ - 3}$

use exponent derivation which yields:

$\displaystyle \frac{d ^{2} y}{d{x}^{2} } = 2 + 42{x}^{ - 4}$

by law of exponent we get:

$\displaystyle \frac{d ^{2} y}{d{x}^{2} } = 2 + \frac{ 42}{ {x}^{4} }$

hence, our answer is d)

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