# How do you find the derivative of 3*(sqrtx) - (sqrtx^3)?

Feb 10, 2016

$\frac{3 \left(1 - x\right)}{2 \sqrt{x}}$

#### Explanation:

Call the function $f \left(x\right)$. Now, write the function using fractional exponents instead of with radicands:

$f \left(x\right) = 3 {x}^{\frac{1}{2}} - {\left({x}^{\frac{1}{2}}\right)}^{3}$

Multiply the exponents in the second term to get:

$f \left(x\right) = 3 {x}^{\frac{1}{2}} - {x}^{\frac{3}{2}}$

Now, each term can be differentiated through the power rule, which states that

$\frac{d}{\mathrm{dx}} \left({x}^{n}\right) = n {x}^{n - 1}$

Recall that constants being multiplied simply stay being multiplied. Applying the power rule to each term gives a derivative of

$f ' \left(x\right) = 3 \left(\frac{1}{2}\right) {x}^{\frac{1}{2} - 1} - \frac{3}{2} {x}^{\frac{3}{2} - 1}$

Simplify.

$f ' \left(x\right) = \frac{3}{2} {x}^{- \frac{1}{2}} - \frac{3}{2} {x}^{\frac{1}{2}}$

While this is a fine final answer, it's often helpful to put functions like this in fractional form so the derivative can be easily set equal to $0$:

$f ' \left(x\right) = \frac{\frac{3}{2} {x}^{- \frac{1}{2}} - \frac{3}{2} {x}^{\frac{1}{2}}}{1} \left({x}^{\frac{1}{2}} / {x}^{\frac{1}{2}}\right)$

$f ' \left(x\right) = \frac{3 - 3 x}{2 {x}^{\frac{1}{2}}}$

$f ' \left(x\right) = \frac{3 \left(1 - x\right)}{2 \sqrt{x}}$