Question

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

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)