Problem 16 - Calculus for Biology and Medicine

Question

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ g(t)=\sqrt{t+\sqrt{t+1}} $$

Step-by-Step Solution

Answer

The derivative is \( g'(t) = \frac{1}{2\sqrt{t + \sqrt{t+1}}} + \frac{1}{4\sqrt{t+1} \cdot \sqrt{t + \sqrt{t+1}}} \).

Step 1: Recognizing the composite functions

The function \( g(t) = \sqrt{t + \sqrt{t+1}} \) is a composite function. The outer function is \( \, f(u) = \sqrt{u} \, \) and the inner function is \( \, u(t) = t + \sqrt{t+1} \). To differentiate \( g(t) \), we will apply the chain rule multiple times.

Step 2: Applying the chain rule for the outer function

Using the chain rule, differentiate the outer function \( f(u) = \sqrt{u} \). Its derivative is \( f'(u) = \frac{1}{2\sqrt{u}} \). Now, replace \( u \) with \( t + \sqrt{t+1} \) to get the derivative of the outer function with respect to \( t \), which is \( \frac{1}{2\sqrt{t + \sqrt{t+1}}} \).

Step 3: Differentiating the inner function

Now differentiate the inner function \( u(t) = t + \sqrt{t+1} \). The derivative of \( u(t) \) with respect to \( t \) is \( u'(t) = 1 + \frac{1}{2\sqrt{t+1}} \), using the chain rule for the square root part.

Step 4: Applying the chain rule to find the derivative of the original function

By combining the results from Steps 2 and 3, apply the chain rule to find the derivative of \( g(t) \). Multiply the derivative of the outer function by the derivative of the inner function: \[ g'(t) = \left( \frac{1}{2\sqrt{t + \sqrt{t+1}}} \right) \cdot \left( 1 + \frac{1}{2\sqrt{t+1}} \right) \].

Step 5: Simplifying the expression

Simplify the expression for the derivative. First expand the multiplication: \[ g'(t) = \frac{1}{2\sqrt{t + \sqrt{t+1}}} + \frac{1}{4\sqrt{t+1} \cdot \sqrt{t + \sqrt{t+1}}} \]. This is the simplified expression for \( g'(t) \).

Key Concepts

Chain Rule

The chain rule is an essential concept in calculus when you are dealing with the differentiation of composite functions. A composite function is formed when one function is nested inside another, which means you need to differentiate the outer function first and then multiply it by the derivative of the inner function. The chain rule makes this process systematic, allowing us to differentiate complex functions efficiently.

When applying the chain rule, always identify:

• The outer function, which operates on the result of the inner function.
• The inner function, which is the argument of the outer function.

For example, consider the function \( g(t) = \sqrt{t + \sqrt{t+1}} \). Here, the outer function is \( f(u) = \sqrt{u} \), and the inner function is \( u(t) = t + \sqrt{t+1} \). By correctly identifying these functions, the chain rule's application becomes straightforward.

Composite Functions

Composite functions occur when you have a function within another function, a common scenario in calculus. Understanding how to work with these is critical, particularly when you need to differentiate them. The process begins with identifying the structure of the composite function.

Let's unpack the function \( g(t) = \sqrt{t + \sqrt{t+1}} \). This function involves nesting because it combines two operations:

• First, the addition of \( t \) to the expression \( \sqrt{t+1} \).
• Secondly, taking the square root of the entire resulting expression.

Thus, the function is built by composing two simpler functions. By recognizing it as a composite function, we can approach its differentiation using the chain rule, which allows us to tackle each part in sequence rather than all at once.

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to an independent variable. When dealing with complex functions, especially those involving multiple layers like composite functions, techniques like the chain rule are vital.

For the function \( g(t) = \sqrt{t + \sqrt{t+1}} \), the differentiation process involves a few steps:

• First, differentiate the outer function, e.g., if the outer function is \( \sqrt{u} \), its derivative is \( \frac{1}{2\sqrt{u}} \).
• Next, differentiate the inner function, which might include additional smaller layers; for instance, \( u(t) = t + \sqrt{t+1} \).

This methodically structured approach to differentiation ensures that even complex nested expressions become manageable by tackling each layer individually.

Derivatives

A derivative is a fundamental concept in calculus that measures how a function changes as its input changes. It is effectively the slope of the tangent line to the function's graph at any given point. Calculating derivatives is crucial for understanding the behavior of functions, such as increasing or decreasing trends and identifying local maxima and minima.

In the context of the function \( g(t) = \sqrt{t + \sqrt{t+1}} \), finding the derivative \( g'(t) \) involves specific steps:

• Apply the chain rule to the composite function, considering both outer and inner derivatives.
• Simplify the resulting expression so that the derivative is clear and concise, like the final form: \( g'(t) = \frac{1}{2\sqrt{t + \sqrt{t+1}}} + \frac{1}{4\sqrt{t+1} \cdot \sqrt{t + \sqrt{t+1}}} \).

Understanding and correctly calculating derivatives allow you to interpret the function's rate of change and various other dynamic properties effectively.