What is the derivative of y = square root of 3x
y = (3x)^.5
dy/dx = .5 (3x)^-.5 * 3
dy/dx = 1.5/sqrt(3x)
y = sqrt ( 3 x ) = sqrt ( 3 ) * sqrt ( x )
d y / d x =sqrt ( 3 ) * 1 / 2 sqrt ( x )
d y / d x = ( 1 / 2 ) * sqrt ( 3 / x )
To find the derivative of \(y = \sqrt{3x}\), we can use the power rule for differentiation. The power rule states that if we have a function of the form \(f(x) = ax^n\), then the derivative of the function is given by \(f'(x) = n \cdot ax^{n-1}\).
In this case, our function is \(y = \sqrt{3x}\). Notice that we can rewrite this as \(y = (3x)^{\frac{1}{2}}\), which is in the form \(ax^n\), with \(a = 3\) and \(n = \frac{1}{2}\).
To find the derivative, we will apply the power rule. According to the power rule, the derivative of \(y = (3x)^{\frac{1}{2}}\) is given by:
\(y' = \frac{1}{2} \cdot (3x)^{\frac{1}{2} - 1}\)
Simplifying this expression, we have:
\(y' = \frac{1}{2} \cdot (3x)^{-\frac{1}{2}}\)
Now, we can simplify this further by expressing it without any negative exponents. Since \(a^{-n} = \frac{1}{a^n}\), we have:
\(y' = \frac{1}{2\sqrt{3x}}\)
Therefore, the derivative of \(y = \sqrt{3x}\) is \(y' = \frac{1}{2\sqrt{3x}}\).
In summary, to find the derivative of \(y = \sqrt{3x}\), we used the power rule for differentiation. First, we rewrote the function in the form \(ax^n\), applied the power rule, and simplified the expression to get the final derivative of \(y' = \frac{1}{2\sqrt{3x}}\).