Problem 36 - Calculus an Applied Approach

Question

Use the General Power Rule to find the derivative of the function. $$ y=2 \sqrt{4-x^{2}} $$

Step-by-Step Solution

Answer

The derivative of the function, \(y=2 \sqrt{4-x^{2}}\), is \(\frac{dy}{dx}=-x(4-x^{2})^{-0.5}\).

Step 1: Rewrite the function

First, transform the function by expressing the square root in terms of a power of 1/2. This gives \(y=2(4-x^{2})^{0.5}\).

Step 2: Apply the General Power Rule and Chain Rule

Next, use the General Power Rule, \(u^n = nu'u^{n-1}\), alongside the Chain Rule, \(\frac{dy}{du} = \frac{dy}{dv} * \frac{dv}{du}\), to differentiate the function. Identify \(u=4-x^{2}\), \(n=0.5\) and hence, \(u’=-2x\). Substitute these and calculate \(\frac{dy}{dx} = 0.5*2*(4-x^{2})^{-0.5}*u’\).

Step 3: Simplify the derivative

Multiply the factors and distribute -x into the function to get \(\frac{dy}{dx}=-x(4-x^{2})^{-0.5}\).

Key Concepts

Chain Rule

The Chain Rule is a fundamental technique in calculus for finding the derivative of a composite function. It helps when a function is nested inside another function. Imagine peeling layers of an onion—each layer represents a function within a function.

The Chain Rule states that if you have a composite function, such as \( f(g(x)) \), its derivative is the derivative of the outer function \( f \) evaluated at the inner function \( g(x) \) times the derivative of the inner function \( g(x) \). This is expressed in the formula:
\[\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}.\]To use the Chain Rule, identify the outer function and the inner function, find their derivatives, and then multiply these derivatives together.

• Outer Function: The function applied last, often requiring the power rule or exponential differentiation.
• Inner Function: The function within, which may be a polynomial or another simple function.

In our example, with \( y = 2(4-x^2)^{0.5} \), the inner function is \( u = 4-x^2 \) and the outer function is \((u)^{0.5}\). First, you differentiate the inner function resulting in \( u' = -2x \). Then you differentiate the power function, which results in \( 0.5u^{-0.5} \). Combining them with the Chain Rule gives the derivative of the original function.

Derivative

A derivative is a measure of how a function changes as its input changes. It tells us the rate of change or the slope of the function at any point. Think of it like the speedometer in a car, showing how fast the function's output is changing.

Mathematically, if you have a function \( y = f(x) \), the derivative is often written as \( \frac{dy}{dx} \). This notation indicates the change in \( y \) with respect to the change in \( x \). Derivatives can predict how a small change in \( x \) will affect \( y \).

• Geometrically, the derivative represents the slope of the tangent line to the curve of the function at any given point.
• The derivative is used for optimization, finding maximum and minimum values of functions.
• It can help analyze the behavior of physical systems and solve real-world problems efficiently.

In solving our problem, the derivative helps find out how the function \( y = 2\sqrt{4-x^2} \) changes as \( x \) changes. By rewriting the function and using the Power and Chain Rules, we could derive the expression for its derivative: \( \frac{dy}{dx} = -x(4-x^2)^{-0.5} \). This captures how the function's output decreases at the square root's rate of change with respect to the input.

Mathematical function

Mathematical functions form the backbone of calculus and are rules that assign a single output to each input from a domain. Functions can be linear, quadratic, polynomial, trigonometric, exponential, or more complex types.

In general, a function is denoted as \( f(x) \), where \( x \) is the input variable, and \( f(x) \) is the output or dependent variable. They are essential in modeling relationships and changes in various fields such as physics, engineering, economics, and everyday life.

• Functions can be represented in different forms such as equations, graphs, or tables.
• A function can describe real-world phenomena, like a car's speed depending on time or population growth over years.
• Functions must have a unique output for each input, respecting the rule of determinability.

In our exercise, the given function \( y = 2 \sqrt{4-x^2} \) is a mathematical function involving a square root. The roots and powers give this function a unique curve and characteristic. By converting it into a power form, we ease the differentiation process to evaluate how it behaves as \( x \) changes. This kind of transformation is standard to help apply calculus principles, like the General Power Rule, effectively.