Problem 28 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

\( h'(t) = \frac{1}{4} \left(4t^4 + \frac{4}{t^4}\right)^{-3/4} \cdot \left(16t^3 - \frac{16}{t^5}\right) \).

Step 1: Identify the Function

We have the function \( h(t) = \left(4t^4 + \frac{4}{t^4}\right)^{1/4} \). This is a composite function with the outer function being a power and the inner function a polynomial.

Step 2: Apply the Chain Rule

Differentiate the function using the chain rule. The chain rule states that if you have a composite function \( f(g(t)) \), the derivative is \( f'(g(t)) \cdot g'(t) \). Here, let \( u = 4t^4 + \frac{4}{t^4} \), so \( h(t) = u^{1/4} \). First, find \( \frac{d}{dt}\left(u^{1/4}\right) = \frac{1}{4}u^{-3/4} \cdot \frac{du}{dt} \).

Step 3: Differentiate the Inner Function

Now, find \( \frac{du}{dt} \). The function \( u = 4t^4 + \frac{4}{t^4} \) can be differentiated term by term:- The derivative of \( 4t^4 \) is \( 16t^3 \).- The derivative of \( \frac{4}{t^4} \) is \( -\frac{16}{t^5} \) (using the power rule). Thus, \( \frac{du}{dt} = 16t^3 - \frac{16}{t^5} \).

Step 4: Substitute and Simplify

Substitute \( \frac{du}{dt} = 16t^3 - \frac{16}{t^5} \) back into the derivative of the outer function: \( h'(t) = \frac{1}{4} \left(4t^4 + \frac{4}{t^4}\right)^{-3/4} \cdot \left(16t^3 - \frac{16}{t^5}\right) \).Then, simplify the expression if possible.

Key Concepts

Understanding the Chain Rule

In calculus, the chain rule is a technique we use to differentiate composite functions. You may wonder what a composite function is. It's a function made by combining two or more functions.
In simple terms, if you have a function nested inside another function, like our example function, you apply the chain rule to find its derivative.

The rule itself states: If you have a composite function \( f(g(t)) \), the derivative of this function is the derivative of \( f \) with respect to \( g(t) \) times the derivative of \( g(t) \) with respect to \( t \).
When you write it mathematically, it looks like this:

• Find \( f'(g(t)) \), the derivative of the outer function.
• Then multiply it with \( g'(t) \), the derivative of the inner function.

This way, you link the changes in the outer function with the changes in the inner function.

In our problem, applying the chain rule separated the outer function, \( u^{1/4} \), and the inner polynomial function, \( 4t^4 + \frac{4}{t^4} \), to find the overall derivative.

Composite Functions Explained

Composite functions sound complex, but let's break it down. Imagine them as a stack of functions lumped together, forming one big process. In our exercise, for instance, we have a composite function \( h(t) = (4t^4 + \frac{4}{t^4})^{1/4} \).

• Here, the inner function is \( g(t) = 4t^4 + \frac{4}{t^4} \).
• The outer function is \( f(u) = u^{1/4} \), where \( u = g(t) \).

To differentiate a composite function, like in our problem, dissect it into its inner and outer parts.
This makes it easier to handle, as you're simplifying the process by breaking it into smaller, manageable parts.

The understanding of composite functions is crucial for applying the chain rule effectively. You can only use the chain rule if you recognize this special form of a function. This concept broadens the range of problems you can tackle, aiding you in both recognizing and solving problems that involve layered dependencies.

The Power Rule in Differentiation

The power rule is one of the simplest yet powerful tools in calculus for finding the derivative of a function. It states that if you have a function \( t^n \), its derivative is \( nt^{n-1} \).
This rule applies directly whenever dealing with powers of t.

For example, if we revisit the inner function \( 4t^4 + \frac{4}{t^4} \) in our exercise, each term here is a polynomial.

• The derivative of \( 4t^4 \) using the power rule is \( 16t^3 \) because you bring down the exponent 4, multiply it by the coefficient 4, and reduce the power to 3.
• Similarly, the derivative of \( \frac{4}{t^4} \) becomes \( -\frac{16}{t^5} \) after applying the power rule. When dealing with negative or fractional exponents, it helps to rewrite them as \( 4t^{-4} \).

In our exercise, the power rule was crucial for differentiating the inner function accurately, which then fed into the chain rule, giving us the full derivative.
The power rule simplifies the differentiation process by breaking complex polynomial parts into manageable pieces.
By mastering it, you're able to handle a wide variety of functions with ease.