Question

What is the rate of increase for the function f(x)= 1/3 (^3 √24)^2x

Answer 1

Answer

You need to simplify the function first until the exponent turns into a plain x. Step 1 Leave 1/3 alone that is the a value, initial value. You are looking for the base

Step 2 Deal with the parenthesis. Factor 24 and you will get 2 and the cube of 3. 

Step 3 Separate the exponent (2) (x)

Step 4 Now square each term inside the parenthesis

2 squared and cube of 3 square  the 2^2 will be 4, the other expression means  cube of 3 times cube of 3 and that's cube of 9

Step 5 Your base should be (4 cube of 9)

Answer 2

Answer:

$f'(x)=\frac{4}{3^{\frac{1}{3}}}$

Step-by-step explanation:

We are given that a function

$f(x)=\frac{1}{3}(\sqrt[3]{24})^2 x$

We have to find the rate of increase of the given function.

Differentiate w.r.t x

Then, we get

$f'(x)=\frac{1}{3}(\sqrt[3]{24})^2$

Using formula :  $\frac{dx^n}{dx}=nx^{n-1}$

$f'(x)=\frac{1}{3}(\sqrt[3]{2\times 2\times 2\times 3})^2$

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

$f'(x)=\frac{4}{3^{1-\frac{2}{3}}}$

Using identity:$\frac{a^x}{a^y}=a^{x-y}$

$f'(x)=\frac{4}{3^{\frac{1}{3}}}$

Hence, the rate of increase of the function =$f'(x)=\frac{4}{3^{\frac{1}{3}}}$