# What is the derivative of #x^(3/2)#?

##### 1 Answer

#### Explanation:

If you have already learnt the concepts of differentiating, skip to the solutions instead.

Differentiation of power functions is found as

By differentiating a function, you are decreasing its power/exponent by 1.

Imagine you are given a *cube* with its corner lengths being

The volume of this cube will be

Thus, let the volume be **dimension** from 3 to 2.

The result is

#V'(x)=3x^2 cm^2#

Notice we differentiate *both* the function and its unit.

Now, its power has decreased to 2, and its unit is now cm *squared* instead of cm *cubed*.

By differentiating, the cube has been reduced from having 3 dimensions to just 2 dimensions.

The cube has been reduced to just a surface of the cube (or plane) ie changed from cube to a square.

If we differentiate again,

#V''(x)=6x cm^1#

Its power is reduced to 1, and reduced to just a *line* from a square.

So what happens when we differentiate

Which is basically nothing (zero).

Try practicing differentiating the volume of a sphere,

**Solution**

#=(3/2)x^(1/2)#

Differentiating *rational* powers of functions ie

Can you imagine what happens by differentiating functions with *irrational* powers ie