differentiate.\ny = 12\\sqrt{x}\n\\frac{d}{dx}(12\\sqrt{x}) = \\square

Explanation:

Step1: Rewrite the function

Rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$, so $y = 12x^{\frac{1}{2}}$.

Step2: Apply the power - rule for differentiation

The power - rule states that if $y = ax^n$, then $y^\prime=\frac{d}{dx}(ax^n)=anx^{n - 1}$. Here $a = 12$ and $n=\frac{1}{2}$. So $\frac{d}{dx}(12x^{\frac{1}{2}})=12\times\frac{1}{2}x^{\frac{1}{2}-1}$.

Step3: Simplify the expression

$12\times\frac{1}{2}x^{\frac{1}{2}-1}=6x^{-\frac{1}{2}}$. Since $x^{-\frac{1}{2}}=\frac{1}{\sqrt{x}}$, the derivative is $\frac{6}{\sqrt{x}}$.

Answer:

$\frac{6}{\sqrt{x}}$