Problem 45 - Calculus for Biology and Medicine

Question

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=2 \sqrt{x^{2}-1} $$

Step-by-Step Solution

Answer

The derivative of \( f(x) = 2\sqrt{x^2 - 1} \) is \( \frac{2x}{\sqrt{x^2 - 1}} \).

Step 1: Identify the function structure

The given function is \( f(x) = 2\sqrt{x^2 - 1} \). Notice that it is a composite function with the form \( g(h(x)) \), where \( g(u) = 2\sqrt{u} \) and \( h(x) = x^2 - 1 \). We will need to apply the chain rule to differentiate this function.

Step 2: Differentiate the outer function

Differentiate the outer function \( g(u) = 2\sqrt{u} \) with respect to \( u \). The derivative \( g'(u) = \frac{d}{du}(2u^{1/2}) = 2 \cdot \frac{1}{2} u^{-1/2} = u^{-1/2} = \frac{1}{\sqrt{u}} \).

Step 3: Differentiate the inner function

Find the derivative of the inner function \( h(x) = x^2 - 1 \) with respect to \( x \). The derivative \( h'(x) = \frac{d}{dx}(x^2 - 1) = 2x \).

Step 4: Apply the chain rule

Use the chain rule, which states \( \frac{d}{dx} f(x) = g'(h(x)) \times h'(x) \). Substituting the derivatives calculated, we have \( \frac{d}{dx} f(x) = \frac{1}{\sqrt{x^2 - 1}} \times 2x \).

Step 5: Simplify the derivative

Simplify the expression from applying the chain rule. The derivative \( \frac{d}{dx} f(x) = \frac{2x}{\sqrt{x^2 - 1}} \).

Key Concepts

Chain Rule

The chain rule is an essential tool in calculus, used to differentiate composite functions. It lets us find the derivative of functions formed by other functions. Suppose we have a function made from two functions like \( g(h(x)) \). To differentiate this, imagine peeling back layers.

• First, compute the derivative of the outer function \( g(u) \), assuming \( u = h(x) \).
• Next, differentiate the inner function \( h(x) \).

Combine these derivatives according to the chain rule:\[\frac{d}{dx} f(x) = g'(h(x)) \cdot h'(x)\]Think of the chain rule as working through paths, starting from the outer layer and moving inward, capturing changes at each level.Deploying this method is powerful for intricate functions where direct differentiation is cumbersome.

Composite Functions

Composite functions join two or more functions into a single expression. They often appear in the structure \( g(h(x)) \), where one function, \( h(x) \), is nested inside another, \( g(u) \). Understanding composite functions is crucial for applying the chain rule efficiently.
A good way to grasp this concept is by:

• Identifying the outer function \( g(u) \), where \( u = h(x) \).
• Then, spotting the inner function \( h(x) \).

Composite functions are common in real-world applications, like calculating the interest compounded over time or the temperature change based on various atmospheric effects.
Recognizing these layers is key to unraveling complex equations and making the differentiation process systematic and straightforward.

Derivative Computation

Computing derivatives involves determining how a function changes as its input changes. For composite functions, translating these changes through multiple layers can seem tricky, but it's manageable with the chain rule.
In derivative computation:

• Identify layers of functions (outer and inner).
• Apply rules: power rule for simple polynomials, product, and quotient rules for attacking more elaborate constructs.
• Simplify at each step for clarity and ease of interpretation.

For the function in our exercise, \( f(x) = 2\sqrt{x^2 - 1} \), we first differentiated \( g(u) = 2\sqrt{u} \) into \( \frac{1}{\sqrt{u}} \), then found the derivative of \( h(x) = x^2 - 1 \) as \( 2x \).
Employing the chain rule, we get \( \frac{2x}{\sqrt{x^2 - 1}} \) as the derivative, illustrating effective derivative computation.