1. Rate Problem

Sodium chlorate crystals are easy to grow in the shape of cubes by allowing a solution of water and sodium chlorate to evaporate slowly. If V is the volume of such a cube with side length x, calculate the derivative when x = 5 mm.

V '(5) = ?

What does this quantity represent?

V '(5) represents the rate at which the volume is increasing with respect to the side length as V reaches 15 mm3.
V '(5) represents the rate at which the volume is increasing as x reaches 15 mm.
V '(5) represents the rate at which the volume is increasing with respect to the side length as x reaches 5 mm.
V '(5) represents the rate at which the side length is increasing with respect to the volume as x reaches 5 mm.
V '(5) represents the volume as the side length reaches 5 mm.

(Attempt)

I dont' know wat to do? Do I need to do the derivatives? Please show me! Thank you!

2. First, we use the formula for the volume of a cube:

$V=x^3.$

We are asked to find the derivative of $V$ with respect to $x$ and evaluate it at $x=5$. To do this, we may either use the Power Rule:

$\frac{d}{dx}x^n=nx^{n-1}$

or we may use the definition of derivative:

$\frac{d}{dx}f(x)=\lim_{\small h\rightarrow 0}\frac{f(x+h)-f(x)}{h}.$

The derivative measures the rate of change of the function with respect to its independent variable. If $V$ is considered a function of $x$, then the derivative of $V$ basically measures the relative change of $V$ when we change $x$ by a very small amount.