Problem 4 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the derivative of each function. \(f(t)=2\left(t^{3}-1\right)^{5}\)

Step-by-Step Solution

Answer

The derivative of the function \(f(t)=2\left(t^{3}-1\right)^{5}\) is: \(f'(t) = 30t^2 (t^{3}-1)^{4}\).

Step 1: Identify Inner and Outer Functions

We are given the function \(f(t) = 2\left(t^{3}-1\right)^{5}\). The inner function is \(g(t) = t^{3}-1\) and the outer function is \(h(u) = 2u^{5}\), where u is the result of the inner function g(t).

Step 2: Find the Derivative of the Inner Function

To find the derivative of the inner function g(t) = t^3 - 1, we use the power rule: \(\frac{d}{dt}(t^3-1) = 3t^2 - 0\). So, the derivative of g(t) is \(g'(t) = 3t^2\).

Step 3: Find the Derivative of the Outer Function

Now we'll find the derivative of the outer function \(h(u) = 2u^5\). Using the power rule, the derivative is \(\frac{d}{du}(2u^{5}) = 10u^4\). So, the derivative of h(u) is \(h'(u) = 10u^4\).

Step 4: Apply the Chain Rule

Now we apply the chain rule to find the derivative of the given function \(f(t)=2\left(t^{3}-1\right)^{5}\): \[f'(t) = h'(g(t)) \cdot g'(t)\] Replace h'(u) with 10u^4 and g'(t) with 3t^2: \[f'(t) = 10(g(t))^4 \cdot 3t^2\] Substitute g(t) back with t^3 - 1: \[f'(t) = 10(t^{3} - 1)^{4} \cdot 3t^2\] Finally, simplify the derivative: \[f'(t) = 30t^2 (t^{3}-1)^{4}\]

Step 5: Write the Final Answer

The derivative of the given function \(f(t)=2\left(t^{3}-1\right)^{5}\) is: \[f'(t) = 30t^2 (t^{3}-1)^{4}\]

Key Concepts

Chain Rule

Understanding the chain rule is crucial when dealing with composite functions. Composite functions are expressions where one function is nested inside another. The chain rule helps us differentiate these kinds of functions. It is essentially a formula that provides a way to deal with the rates of change of composite functions.

The chain rule states that if you have a function composed of an outer function and an inner function, like this:

• The derivative of the composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

For a function \( f(x) = h(g(x)) \), the chain rule formula can be expressed as: \[ f'(x) = h'(g(x)) \cdot g'(x) \]

In applying the chain rule, always ensure that you completely differentiate each "layer" of functions involved, following the order from the outer function down to the inner function.

Power Rule

The power rule is one of the simplest and most widely used rules for differentiating functions. It applies specifically to functions of the form \(x^n\), where \(n\) is any real number. The power rule states:

• If \( f(x) = x^n \), then its derivative \( f'(x) = nx^{n-1} \).

This rule makes it straightforward to find derivatives of polynomial expressions. It allows you to quickly calculate the rate of change for functions with variable exponents.

For example, when using the power rule to differentiate \(t^3\), you bring down the 3 as a coefficient and reduce the power by one:
\( \frac{d}{dt}(t^3) = 3t^2 \).

The power rule is applied both to the inner and outer functions in our case, showing its flexibility and foundational role in calculus.

Outer Function

When dealing with composite functions, the outer function is the function that "encapsulates" the inner function. For example, in the problem \(f(t) = 2(t^3 - 1)^5\), the outer function can be seen as \(h(u) = 2u^5\), where \(u = t^3 - 1\).

It's important to understand the outer function's role in determining the overall behavior of the composite function.

• Differentiate the outer function by treating the inner function as a variable.
• In our example, take the derivative of \(h(u) = 2u^5\) obtaining \(h'(u) = 10u^4\).

Outer functions typically influence the "shape" of the graph, reflecting how changes in the inner function affect the composite structure overall. Being precise in this differentiation step sets a solid foundation when applying the chain rule.

Inner Function

The inner function is the core part of a composite function that is nested within another function. In this exercise, the inner function is \(g(t) = t^3 - 1\).

Identifying the inner function is the first step in using the chain rule effectively. It is crucial because it dictates how the outer function will be differentiated.

• Differentiate the inner function independently. Here, \(g'(t) = 3t^2\) is found using the power rule.
• Recognize the inner function as the component affected first when input changes.

Understanding the inner function’s role helps trace how each part of the composite function contributes to the overall behavior and rate of change. Properly differentiating the inner function is essential for obtaining the correct final derivative.