Rose Martin
03/24/2024 · Junior High School
Differentiate. \[ y=12 \sqrt{x} \]
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Step-by-step Solution
To differentiate the function \( y = 12 \sqrt{x} \), we can first rewrite the square root in exponent form:
\[
y = 12 x^{1/2}
\]
Now, we can apply the power rule for differentiation, which states that if \( y = k x^n \), then \( \frac{dy}{dx} = k n x^{n-1} \).
In this case, \( k = 12 \) and \( n = \frac{1}{2} \). Applying the power rule:
\[
\frac{dy}{dx} = 12 \cdot \frac{1}{2} x^{\frac{1}{2} - 1}
\]
Calculating this gives:
\[
\frac{dy}{dx} = 12 \cdot \frac{1}{2} x^{-\frac{1}{2}} = 6 x^{-\frac{1}{2}}
\]
We can also rewrite \( x^{-\frac{1}{2}} \) as \( \frac{1}{\sqrt{x}} \):
\[
\frac{dy}{dx} = \frac{6}{\sqrt{x}}
\]
Thus, the derivative of \( y = 12 \sqrt{x} \) is:
\[
\frac{dy}{dx} = \frac{6}{\sqrt{x}}
\]
Quick Answer
The derivative of \( y = 12 \sqrt{x} \) is \( \frac{6}{\sqrt{x}} \).
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