Sure, let's find the derivative of the function step by step.
Given the function:
[tex]\[ f(t) = (6t + 1)^{\frac{2}{3}} \][/tex]
We need to find the derivative [tex]\( f'(t) \)[/tex].
### Step 1: Identify the Components
Notice that the function can be written in the form of a power of [tex]\(u\)[/tex], where [tex]\( u = 6t + 1 \)[/tex]. So we can rewrite:
[tex]\[ f(t) = u^{\frac{2}{3}} \][/tex]
### Step 2: Use the Chain Rule
The chain rule states that if you have a composite function [tex]\(f(g(t))\)[/tex], then the derivative is:
[tex]\[ \frac{d}{dt} f(g(t)) = f'(g(t)) \cdot g'(t) \][/tex]
In our case, let:
[tex]\[ g(t) = 6t + 1 \][/tex]
So:
[tex]\[ f(g) = g^{\frac{2}{3}} \][/tex]
Hence, we need to find:
[tex]\[ \frac{d}{dt} (g^{\frac{2}{3}}) \][/tex]
### Step 3: Differentiate the Outer Function
To differentiate the outer function [tex]\( g^{\frac{2}{3}} \)[/tex]:
[tex]\[ \frac{d}{dg} g^{\frac{2}{3}} = \frac{2}{3} g^{-\frac{1}{3}} \][/tex]
### Step 4: Differentiate the Inner Function
The inner function is [tex]\( g(t) = 6t + 1 \)[/tex]:
[tex]\[ \frac{d}{dt} (6t + 1) = 6 \][/tex]
### Step 5: Combine the Results Using the Chain Rule
Combining these results using the chain rule, we get:
[tex]\[ \frac{d}{dt} (6t + 1)^{\frac{2}{3}} = \frac{2}{3} (6t + 1)^{-\frac{1}{3}} \cdot 6 \][/tex]
### Step 6: Simplify the Expression
Simplify the expression:
[tex]\[ \frac{d}{dt} (6t + 1)^{\frac{2}{3}} = \frac{2 \cdot 6}{3} (6t + 1)^{-\frac{1}{3}} = 4 (6t + 1)^{-\frac{1}{3}} \][/tex]
So the derivative of the function is:
[tex]\[ f'(t) = \frac{4}{(6t + 1)^\frac{1}{3}} \][/tex]
Hence, the final answer is:
[tex]\[ f'(t) = \frac{4}{(6t + 1)^{0.333333333333333}} \][/tex]
