Problem 26 - Calculus for Biology and Medicine

Question

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ h(t)=\left(t^{4}-5 t\right)^{5 / 2} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Outer and Inner Functions

The given function is a composite function of the form \( h(t) = (u(t))^{5/2} \) where \( u(t) = t^4 - 5t \). Here, the outer function is \( (u)^{5/2} \) and the inner function is \( t^4 - 5t \).

Step 2: Apply the Chain Rule

To differentiate a composite function \( h(t) = (u(t))^{5/2} \), apply the chain rule: \( \frac{d}{dt} h(t) = \frac{d}{du} (u)^{5/2} \cdot \frac{d}{dt} u(t) \). This requires differentiating both the outer and inner functions.

Step 3: Differentiate the Outer Function

The outer function is \( (u)^{5/2} \). Differentiate with respect to \( u \) to get \( \frac{d}{du} (u)^{5/2} = \frac{5}{2} u^{3/2} \).

Step 4: Differentiate the Inner Function

The inner function is \( u(t) = t^4 - 5t \). Differentiate with respect to \( t \) to get \( \frac{d}{dt} (t^4 - 5t) = 4t^3 - 5 \).

Step 5: Combine the Results

Substitute the derivatives back into the chain rule expression: \( \frac{d}{dt} h(t) = \frac{5}{2} (t^4 - 5t)^{3/2} \cdot (4t^3 - 5) \).

Step 6: Simplify the Expression

The expression \( \frac{d}{dt} h(t) = \frac{5}{2} (t^4 - 5t)^{3/2} (4t^3 - 5) \) is the derivative of the given function with respect to \( t \).

Step 7: Final Verification

Verify each step to ensure all derivatives were calculated correctly and the chain rule was applied properly. The final simplified derivative is correct.

Key Concepts

Chain Rule

The chain rule is an essential tool in calculus differentiation. It's used when differentiating composite functions, which are functions composed of two or more functions. Essentially, it provides a method for finding the derivative of a function within a function.

The core idea behind the chain rule is that you take the derivative of the outer function and multiply it by the derivative of the inner function. This approach ensures that all layers of functions are considered, providing a complete understanding of the rate of change.

To recall, for a function like \( h(t) = (u(t))^{5/2} \), the chain rule tells us to differentiate as follows:

• Outer function derivative: \( \frac{d}{du} (u(t))^{5/2} = \frac{5}{2} u^{3/2} \)
• Inner function derivative: \( \frac{d}{dt} (t^4 - 5t) = 4t^3 - 5 \)

By using the chain rule, each part retains its relationship to the whole function, ensuring that the derivative is accurate and thorough.

Composite Function

A composite function is formed when one function is applied to the result of another function. It is expressed in the form \( h(t) = f(g(t)) \). In our problem, \( h(t) = (t^4 - 5t)^{5/2} \), we identify it as a composite function where:

• **Inner function** \( g(t) = t^4 - 5t \)
• **Outer function** \( f(u) = u^{5/2} \), where \( u = g(t) \)

The complexity of a composite function arises from having multiple layers. You have to understand the role of each layer to accurately differentiate and analyze the behavior of the whole function.

Each component carries specific information about the function's overall structure and how changes in the input variable \( t \) affect the output. Mastering this concept helps in navigating problems that involve intricate functions.

Derivative Calculation

Calculating the derivative of a function is finding the rate at which the function's output changes with respect to changes in the input. This calculation is pivotal in many fields that require understanding dynamic systems.

Here are the steps followed for derivative calculation in this example:

• Identify and differentiate the inner function: \( \frac{d}{dt} (t^4 - 5t) = 4t^3 - 5 \).
• Determine and differentiate the outer function: \( \frac{d}{du} (u)^{5/2} = \frac{5}{2} u^{3/2} \).
• Apply the chain rule by multiplying these derivatives: \( \frac{d}{dt} h(t) = \frac{5}{2} (t^4 - 5t)^{3/2} \cdot (4t^3 - 5) \).

This careful method ensures that all aspects of the function are properly addressed, thus providing an accurate result. Re-checking each step ensures no errors in the mathematical process, which emphasizes the importance of meticulous calculation.