How do I find the derivative of $q=\sqrt{15r-r^3}$? What is the rule for derivatives of a radical?

Question

My equation is $q=\sqrt{15r-r^3}$
I need to find the derivative, but do not know the rule for finding the derivative of a radical.
Can some one give me a step-by-step example of how to solve a similar problem and explain how to take the derivative of a radical and how to know what rules I use?

Answer 1

$q = (15r - r^3)^{\frac{1}{2}}$

Using chain rule,

$\frac{dq}{dr} = \frac{1}{2}(15r-r^3)^{-\frac{1}{2}}(15-3r^2)$

Tidying this up gives you

$\frac{15-3r^2}{2\sqrt{15r-r^3}}$

Answer 2

there are two ways:

1.) $$(\sqrt{x})' = \frac{1}{2\sqrt{x}}$$

2.) $$(\sqrt{x})' = (x^{\frac{1}{2}})' = \frac{1}{2}x^{-\frac{1}{2}}$$

both will work, but in your case I would write it like:

$$((15r-r^3)^{\frac{1}{2}})'$$

and then find

$$\frac{dq}{dr} = \frac{1}{2}(15r-r^3)^{-\frac{1}{2}}\cdot(15-3r^2)$$

but I guess you should remember derivatives' table by heart :)