Problem 30 - Calculus for Biology and Medicine

Question

Differentiate $$ f(x)=\sqrt{a x^{2}-2} $$ with respect to \(x\). Assume that \(a\) is a positive constant.

Step-by-Step Solution

Answer

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

Step 1: Identify the Differentiation Rule

The function given is \( f(x) = \sqrt{a x^{2}-2} \). To differentiate this, notice that it involves a composition of functions: the outer function being the square root and the inner function being a quadratic expression. We'll use the chain rule for differentiation, which states that if \( g(x) = h(k(x)) \), then \( g'(x) = h'(k(x)) \cdot k'(x) \).

Step 2: Differentiate the Outer Function

Let \( u = ax^2 - 2 \). The outer function is \( \sqrt{u} = u^{1/2} \). The derivative of \( u^{1/2} \) with respect to \( u \) is \( \frac{1}{2} u^{-1/2} \).

Step 3: Differentiate the Inner Function

For the inner function, \( u = ax^2 - 2 \), differentiate it with respect to \( x \). The derivative of \( ax^2 - 2 \) with respect to \( x \) is \( 2ax \).

Step 4: Apply the Chain Rule

Using the chain rule, combine the derivatives. The derivative of \( f(x) \) is: \[ f'(x) = \frac{1}{2} (ax^2 - 2)^{-1/2} \cdot (2ax) \]

Step 5: Simplify the Expression

Simplify the derivative expression obtained in Step 4:\[ f'(x) = \frac{1}{2} \cdot 2ax \cdot (ax^2 - 2)^{-1/2} = \frac{ax}{\sqrt{ax^2 - 2}} \]

Key Concepts

Differentiation

Differentiation is a fundamental operation in calculus that allows us to find the rate at which a function changes. This process provides us with the derivative of a function. At its essence, differentiation lets us understand how sensitive a function is to changes in its input values, which can be pivotal for both theoretical and practical applications. In the context of our exercise, we are differentiating a function that involves a square root of a quadratic expression.

When faced with any differentiation problem, it's crucial to recognize the type of function you're working with and the appropriate rule to apply. In more complex scenarios, like dealing with compositions of functions, particular strategies such as the chain rule become essential.

• Identify the type of function.
• Determine the rules required for differentiation.
• Apply the chain or product rule as needed.

Mastering differentiation requires practice, but it's all about recognizing patterns and applying the right techniques.

Chain Rule

The chain rule is a powerful technique for differentiation, especially useful when dealing with composite functions, where one function is nested within another. It's essentially the rule which helps us "chain" the derivatives of inner and outer functions to get the derivative of the entire composition. In simpler terms, it allows us to take derivatives of functions that have other functions within them.

In the given exercise, the chain rule is the key tool because our function involves:

• An outer function: the square root function, which can be written as \( u^{1/2} \).
• An inner function: a quadratic polynomial \( ax^2 - 2 \).

To apply the chain rule, follow these steps:

• Differentiate the outer function with respect to the inner function.
• Differentiate the inner function with respect to the variable, like x in our problem.
• Multiply these derivatives together to find the derivative of the entire composite function.

This method allows us to systematically address complex functions by breaking them down into simpler parts.

Derivative

A derivative represents the rate of change of a function with respect to its variable. Conceptually, you can think of it as the slope of the tangent line at any given point on a function's graph. Calculating derivatives is a foundational skill in calculus, allowing us to explore the behavior of functions beyond their values.

In this exercise, finding the derivative of the function \( f(x)=\sqrt{ax^2-2}\) involves some straightforward yet critical steps. First, recognize that taking the derivative of a simple function like \( x^2 \) results in \( 2x \). Similarly, with differentials, derivatives provide us insights on how functions grow or shrink.

In more complex functions, derivatives help us understand:

• Instantaneous rates of change.
• Critical points, which can indicate maxima or minima.
• Concavity and convexity, giving insights into the curvature of function graphs.

Mastering derivatives can enhance your understanding of both theoretical mathematics and practical applications like physics, engineering, and economics.

Composite Functions

Composite functions involve one function nested within another, a concept where one output becomes the input for the next function. These are prevalent in calculus and demand special attention during differentiation. To decompose a composite function:

• Identify the outer function - such as the square root in the expression \(\sqrt{ax^2 - 2}\).
• Determine the inner function - in this case, the polynomial \(ax^2 - 2\).

The process of dealing with such functions requires careful application of the chain rule to find derivatives accurately. Understanding composite functions is crucial, as they arise in various mathematical models and real-world applications.

When breaking down such functions, always:

• Keep track of what functions are composed and in which order.
• Apply the chain rule appropriately, as missing this step can lead to incorrect derivatives.
• Verify results by simplifying the derivative expression.

By mastering composite functions, learners can tackle more intricate problems, expanding their calculus toolkit. These skills are invaluable across many disciplines, offering new ways to model and solve complex problems.