7 multiple choice 1 point what is the derivative of ( 5 sqrt { x } ) ? ( 5 x ^ { - 1 / 2 } ) ( \frac { 5 } {…

7 multiple choice 1 point what is the derivative of ( 5 sqrt { x } ) ? ( 5 x ^ { - 1 / 2 } ) ( \frac { 5 } { 2 } x ^ { 1 / 2 } ) ( 10 x ^ { 3 / 2 } ) ( \frac { 5 } { 2 } x ^ { - 1 / 2 } ) ( \frac { 5 } { 2 } x ^ { 3 / 2 } ) ( 10 x ^ { - 1 / 2 } ) ( 5 x ^ { 1 / 2 } ) clear my selection

7 multiple choice 1 point what is the derivative of ( 5 sqrt { x } ) ? ( 5 x ^ { - 1 / 2 } ) ( \frac { 5 } { 2 } x ^ { 1 / 2 } ) ( 10 x ^ { 3 / 2 } ) ( \frac { 5 } { 2 } x ^ { - 1 / 2 } ) ( \frac { 5 } { 2 } x ^ { 3 / 2 } ) ( 10 x ^ { - 1 / 2 } ) ( 5 x ^ { 1 / 2 } ) clear my selection

Answer

Explanation:

Step1: Rewrite the function

Rewrite (5\sqrt{x}) as (5x^{\frac{1}{2}}).

Step2: Apply the power rule

The power rule for differentiation is ((x^n)^\prime = nx^{n - 1}). For (y = 5x^{\frac{1}{2}}), using the constant - multiple rule ((cf(x))^\prime=c f^\prime(x)) (where (c = 5) and (f(x)=x^{\frac{1}{2}})), we have (y^\prime=5\times\frac{1}{2}x^{\frac{1}{2}-1}).

Step3: Simplify the exponent

(\frac{1}{2}-1=-\frac{1}{2}), so (y^\prime=\frac{5}{2}x^{-\frac{1}{2}}).

Answer:

(\frac{5}{2}x^{-1/2}) (the fourth option)