Problem 23

Question

In Problems 21-28, find the indicated derivative. $$ D_{t}\left(\frac{3 t-2}{t+5}\right)^{3} $$

Step-by-Step Solution

Verified

Answer

The derivative is $ \frac{51(3t-2)^2}{(t+5)^4} $.

1Step 1: Define the Function

The function given is $ \left(\frac{3t-2}{t+5}\right)^3 $. Our goal is to find its derivative with respect to $ t $, denoted as $ D_t( \left(\frac{3t-2}{t+5}\right)^3 ) $.

2Step 2: Apply the Chain Rule

To differentiate $ \left(\frac{3t-2}{t+5}\right)^3 $, we use the chain rule. The chain rule states that the derivative of $ f(g(t)) $ is $ f'(g(t)) \cdot g'(t) $. Here, $ f(u) = u^3 $ and $ u = \frac{3t-2}{t+5} $.

3Step 3: Differentiate the Outer Function

First, differentiate the outer function $ f(u) = u^3 $ with respect to $ u $, which gives us $ 3u^2 $. Thus, $ f'(u) = 3\left(\frac{3t-2}{t+5}\right)^2 $.

4Step 4: Differentiate the Inner Function

Now differentiate the inner function $ u = \frac{3t-2}{t+5} $. This is a quotient and requires the quotient rule. The quotient rule is $ \frac{d}{dt}\left(\frac{a}{b}\right) = \frac{a'b - ab'}{b^2} $.

5Step 5: Apply the Quotient Rule

Let $ a = 3t - 2 $ and $ b = t + 5 $. Then $ a' = 3 $ and $ b' = 1 $. The quotient rule gives:\[ g'(t) = \frac{(3)(t+5) - (3t-2)(1)}{(t+5)^2} = \frac{3t + 15 - 3t + 2}{(t+5)^2} = \frac{17}{(t+5)^2} \].

6Step 6: Combine Using the Chain Rule

Substitute the results from Steps 3 and 5 into the chain rule:\[ D_t\left(\left(\frac{3t-2}{t+5}\right)^3\right) = 3\left(\frac{3t-2}{t+5}\right)^2 \cdot \frac{17}{(t+5)^2} \].

7Step 7: Simplify the Expression

Combine and simplify the expression:\[ D_t\left(\left(\frac{3t-2}{t+5}\right)^3\right) = \frac{51(3t-2)^2}{(t+5)^4} \].

Key Concepts

Chain RuleQuotient RuleDifferentiation Steps

Chain Rule

The chain rule is an essential technique in calculus used for finding the derivative of composite functions, that is, functions within functions. If you have a function of the form $ f(g(t)) $, the chain rule allows you to differentiate it by taking the derivative of the outer function and multiplying it by the derivative of the inner function.
For example, in our exercise, the function is $ \left(\frac{3t-2}{t+5}\right)^3 $. Here, the outer function is $ f(u) = u^3 $ and the inner function is $ u = \frac{3t-2}{t+5} $.
This means we first differentiate $ f(u) $ with respect to $ u $, then $ u $ with respect to $ t $, and finally multiply these derivatives together to get the complete derivative using the chain rule. Remember,

Apply the chain rule to differentiate composite functions effectively.
The chain rule can also be expressed as: $ f'(g(t)) \cdot g'(t) $.
Each component's derivative needs to be computed during this process.

Quotient Rule

The quotient rule is another differentiation technique crucial whenever you need to find the derivative of a function that involves division, specifically a quotient of two functions. If your function is of the form $ \frac{a(t)}{b(t)} $, the quotient rule states the derivative is:
\[\frac{d}{dt}\left(\frac{a}{b}\right) = \frac{a'b - ab'}{b^2}\]
In this exercise, our inner function $ u = \frac{3t-2}{t+5} $ presents itself as a quotient and, accordingly, requires applying the quotient rule.
You identify:

$ a(t) = 3t - 2 $
$ b(t) = t + 5 $
The derivatives: $ a'(t) = 3 $ and $ b'(t) = 1 $.

By substituting these into the quotient rule formula, we find $ g'(t) = \frac{17}{(t+5)^2} $. This derivative is then used in our chain rule process.

Differentiation Steps

When faced with a complex differentiation problem, breaking it down into manageable steps helps. Here's how we tackled the exercise:
**Step-by-Step Explanation:**
1. **Identify the Function Type:** Recognize $ \left(\frac{3t-2}{t+5}\right)^3 $ as a composite function requiring the chain rule.
2. **Differentiate the Outer Function:** Apply the chain rule by differentiating $ u^3 $ to get $ 3u^2 $. Substitute $ u = \frac{3t-2}{t+5} $ back, leading to $ 3\left(\frac{3t-2}{t+5}\right)^2 $.
3. **Differentiate the Inner Function:** Use the quotient rule to differentiate $ \frac{3t-2}{t+5} $ which results in $ g'(t) = \frac{17}{(t+5)^2} $.
4. **Combine Using the Chain Rule:** Multiply the results of the above differentiations together: