Problem 21 - Calculus Single Variable

Question

Use the Chain Rule-Power Rule to differentiate the given expression with respect to \(x\). $$ \sqrt{4+x^{2}} $$

Step-by-Step Solution

Answer

The derivative is \( \frac{x}{\sqrt{4+x^2}} \).

Step 1: Rewrite the Expression

First, rewrite the expression \( \sqrt{4+x^{2}} \) using the power rule notation. The square root can be expressed as an exponent of \( \frac{1}{2} \):\[(4 + x^2)^{\frac{1}{2}}\]

Step 2: Identify Inner and Outer Functions

Identify the inner function \( u(x) = 4 + x^2 \) and the outer function \( v(u) = u^{\frac{1}{2}} \). The problem requires the use of the chain rule, which involves differentiating these separately.

Step 3: Differentiate Inner Function

Differentiate the inner function \( u(x) = 4 + x^2 \) with respect to \( x \). The derivative of \( u \) is:\[\frac{du}{dx} = \frac{d}{dx}(4 + x^2) = 0 + 2x = 2x\]

Step 4: Differentiate Outer Function

Differentiate the outer function \( v(u) = u^{\frac{1}{2}} \) with respect to its argument \( u \). Using the power rule, the derivative is:\[\frac{dv}{du} = \frac{1}{2}u^{-\frac{1}{2}}\]

Step 5: Apply Chain Rule

Apply the chain rule to find the derivative of the original function. The chain rule states:\[\frac{dy}{dx} = \frac{dv}{du} \times \frac{du}{dx}\]Substitute the derivatives from Steps 3 and 4:\[\frac{dy}{dx} = \left( \frac{1}{2}(4+x^2)^{-\frac{1}{2}} \right) \times (2x)\]Simplify the expression:\[\frac{dy}{dx} = \frac{x}{\sqrt{4+x^2}}\]

Step 6: Final Result

The derivative of the function \( \sqrt{4 + x^2} \) with respect to \( x \) is:\[\frac{dy}{dx} = \frac{x}{\sqrt{4+x^2}}\]

Key Concepts

Derivative

A derivative represents how a function changes as its input changes. It's a core concept in calculus and means finding the rate at which one quantity changes with respect to another. The derivative is essentially a measure of how a function's value shifts with minute changes in input value. For example, if you have a function that describes the position of a car over time, the derivative of that function would show the car's velocity at any given point in time.

• To compute a derivative, we use various rules and operations, such as the chain rule.
• Derivatives offer insight into the function's behavior, like increasing or decreasing trends, maxima, and minima.

In our exercise, the derivative provides us with the rate of change of the expression \( \sqrt{4+x^{2}} \) with respect to \( x \), showing how it changes as \( x \) increases or decreases.

Power Rule

The power rule is one of the simplest and most frequently used rules for differentiation. It helps calculate the derivative of functions in the form of \( x^n \), where \( n \) is any real number. According to the power rule, if you have a function \( f(x) = x^n \), its derivative \( f'(x) \) is \( nx^{n-1} \).

• The rule simplifies the process of finding derivatives for polynomial functions.
• It applies to positive, negative, and fractional powers alike.

In the given exercise, after rewriting the square root expression as \((4 + x^2)^{1/2}\), the power rule is applied to both the inner and outer functions, necessary for implementing the chain rule effectively.

Differentiation

Differentiation is the process of finding a derivative of a function. It involves several rules, with each designed to handle different forms of mathematical expressions. Differentiation is essential for understanding relationships between variables, especially in scientific and economic studies.

• Differentiation helps identify slopes of curves or instantaneous rates of change.
• It allows us to understand how functions behave and interact in changing situations.
• For complex functions, techniques like the chain rule become pivotal.

In our context, differentiating \( \sqrt{4+x^2}\) involves several steps, including rewriting the function and differentiating both inner and outer components, illustrating the power of composite functions in calculus.

Inner Function

An inner function is part of a composite function. Here, one function is nested inside another. To differentiate a function like \((4+x^{2})^{1/2}\), we identify \(4+x^2\) as the inner function. In chain rule terminology, this is typically labeled \(u(x)\).

• The inner function influences the overall behavior of the composite function.
• It is crucial to differentiate the inner function separately before applying the chain rule.

In our example, differentiating the inner function \(u(x) = 4 + x^2\) gives us \(du/dx = 2x\), preparing us to apply the chain rule by combining it with the derivative of the outer function.

Outer Function

The outer function is the main structure that contains another function within it, forming a composite. In our exercise, \(u^{1/2}\) is considered the outer function, given as \(v(u) = u^{1/2}\). Understanding the outer function is essential for using the chain rule correctly, as it shows how the inner function's output is manipulated.

• The outer function often modifies the whole expression, like applying a power or exponential form.
• To differentiate it, we must first express it in a differentiable form, often using rules like the power rule.

For instance, the derivative of \(v(u) = u^{1/2}\), with respect to \(u\) is found using the power rule: \(dv/du = 1/2u^{-1/2}\). Combining this with the inner derivative via the chain rule, gives the derivative of the composite function.