Question

Compute the derivative of the given function. $$f(t)=(3 t-2)^{5}$$

Short answer

The derivative is \( f'(t) = 15(3t - 2)^4 \).

Step-by-step solution

Identify the form of the function

The given function is \( f(t) = (3t - 2)^5 \). This is a composite function, which can be solved using the chain rule for derivatives.

Apply the chain rule

The chain rule states that the derivative of a composite function \( g(h(t)) \) is \( g'(h(t)) \, h'(t) \). Here, consider \( g(u) = u^5 \) and \( u = 3t - 2 \). We will differentiate these separately.

Differentiate the outer function

Differentiate \( g(u) = u^5 \) with respect to \( u \). This gives \( g'(u) = 5u^4 \).

Differentiate the inner function

Differentiate \( u = 3t - 2 \) with respect to \( t \). This gives \( u' = 3 \).

Combine derivatives using the chain rule

Substitute back to find \( f'(t) \). So, \( f'(t) = g'(u) \, h'(t) = 5(3t - 2)^4 \, \cdot \, 3 \).

Simplify the expression

Simplify the expression to obtain the final answer: \( f'(t) = 15(3t - 2)^4 \).

Key concepts

Chain Rule

The chain rule is a fundamental technique in calculus for finding the derivative of a composite function. It allows us to differentiate complex functions that can be broken down into simpler, nested functions. In essence, the chain rule states that if you have a function that is composed of two functions, say \( g(h(t)) \), the derivative \( f'(t) \) is found by differentiating the outer function \( g \) with respect to \( h(t) \) and then multiplying by the derivative of the inner function \( h(t) \).
This can be expressed as:

\[ f'(t) = g'(h(t)) \, h'(t) \]

In our specific problem, \( g(u) = u^5 \) and \( h(t) = 3t - 2 \). These functions are differentiated separately first and then combined using the chain rule. Applying this rule correctly is crucial for handling derivatives of composite functions.

Composite Function

A composite function is a function formed by combining two or more functions. It's like nesting one function inside another. For example, \( f(t) = (3t - 2)^5 \) is a composite function where the inner function \( h(t) = 3t - 2 \) is nested inside the outer function \( g(u) = u^5 \).
Here's how you can recognize and differentiate a composite function:

• Identify the outer and inner functions in the composition. In this exercise, \( g \) is the outer and \( h \) is the inner.
• Understand that to differentiate this type of function, both parts must be addressed: the outer function's effect and how the inner function changes with respect to its variable.

This structured process allows you to systematically take on challenging derivatives that at first seem intimidating. By focusing on each function separately, you can manage and simplify the differentiation process.

Differentiation

Differentiation is the process of finding the derivative of a function, which measures how the function's value changes as its input changes. It's like finding the slope of a function at any given point, representing the rate of change.
When differentiating composite functions, the chain rule is frequently applied to handle the inner and outer functions.

• First, differentiate the outer function with respect to its inner function.
• Then, differentiate the inner function with respect to its variable.

In our case:

• Differentiate \( g(u) = u^5 \) to get \( g'(u) = 5u^4 \).
• Differentiate \( u = 3t - 2 \) to get \( u' = 3 \).

These derivatives are then combined to find the overall derivative using the chain rule.

Simplifying Expressions

Simplifying expressions is the final step in calculus problems, ensuring the derivative is as straightforward and concise as possible. Once you've differentiated using the chain rule, you'll often end up with an expression that can be simplified for clarity and ease of use.
In our problem, after applying the chain rule, the expression was:

\[ f'(t) = 5(3t - 2)^4 \, \cdot \, 3 \]

To simplify, multiply the constants and keep the expression neat. Thus, we have:

\[ f'(t) = 15(3t - 2)^4 \]

Simplifying doesn't change the value; it just makes the derivative easier to read and understand. This step is crucial to ensure that you can quickly interpret and use the derivative in further analysis or application.