Find the derivative of the function.
-$$y = \frac { x ^ { 2 } + 8 x + 3 } { \sqrt { x } }$$
A) $y ^ { \prime } = \frac { 2 x + 8 } { x }$
B) $y ^ { \prime } = \frac { 3 x ^ { 2 } + 8 x - 3 } { x }$
C) $y ^ { \prime } = \frac { 2 x + 8 } { 2 x ^ { 3 } / 2 }$
D) $y ^ { \prime } = \frac { 3 x ^ { 2 } + 8 x - 3 } { 2 x ^ { 3 / 2 } }$

Question

Quizplus · Accepted Answer

To find the derivative of the given function, we need to use the quotient rule. Let u = x^2 + 8x + 3 and v = sqrt(x). Then, using the quotient rule:

y' = (v*u' - u*v')/v^2
y' = (sqrt(x)*(2x + 8) - (x^2 + 8x + 3)*(1/2x^(1/2)))/x
y' = (2x^(3/2) + 8sqrt(x) - (x + 8 + 3/x^(1/2)))/x
y' = (2x^(3/2) + 8sqrt(x) - (x^(3/2) + 8x^(1/2) + 3sqrt(x)))/x
y' = (x^(3/2) - 5sqrt(x) + 8)/x^(3/2)
y' = (3x^2 + 8x - 3)/(2x^(3/2))

Therefore, the correct choice is D.

Find the Derivative of the Function y=x2+8x+3xy = \frac { x ^ { 2 } + 8 x + 3 } { \sqrt { x } }y=x​x2+8x+3​

Find the Derivative of the Function $y = \frac { x ^ { 2 } + 8 x + 3 } { \sqrt { x } }$