Problem 38 - Calculus an Applied Approach

Question

Use the General Power Rule to find the derivative of the function. $$ f(x)=\left(25+x^{2}\right)^{-1 / 2} $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x)=(25+x^{2})^{-1 / 2}\) is \(-x(25+x^{2})^{-3/2}\)

Step 1: Identify the Inner Function

First, we identify the inner function of the form \(g(x)\). Here, our \(g(x) = 25+x^{2}\)

Step 2: Calculate the Derivative of the Inner Function

Calculate \(g'(x)\) by taking the derivative of \(g(x) = 25+x^{2}\). The derivative of a constant (25) is zero and the power rule gives the derivative of \(x^{2}\) to be \(2x\). Thus, \(g'(x) = 2x\)

Step 3: Apply the General Power Rule

The General Power Rule states that the derivative of \(g(x)^n\) is \(n*g(x)^{n-1}*g'(x)\). Substituting \(g(x) = 25+x^{2}\), \(n = -1/2\), and \(g'(x) = 2x\), we get the derivative to be \(-1/2*(25+x^{2})^{-3/2}*2x\)

Step 4: Simplify the Derivative

Simplify the derivative: \(-x(25+x^{2})^{-3/2}\)

Key Concepts

Understanding Derivatives

In calculus, a derivative represents the rate at which a function changes at any given point. Simply put, it tells us how the output of a function responds to changes in the input. Understanding derivatives is crucial because it serves as the foundation for other concepts like optimization and motion.

The process of finding a derivative is known as differentiation. When we differentiate a function, we essentially are taking the slope of the function's tangent line at any given point.

• For constant functions, derivatives are zero. This is because constants don't change, so their rate of change is zero.
• For linear functions like \(f(x) = ax + b\), the derivative will always be \(a\) as it's the constant rate of change.

In our specific problem, we aim to find the derivative of a more complex function \(f(x)=(25+x^2)^{-1/2}\). The goal is to determine how rapidly this function changes with respect to \(x\).

Introduction to the Power Rule

The power rule is a fundamental tool in calculus used when finding derivatives of functions of the form \(x^n\). The rule states that if we have a function \(f(x) = x^n\), the derivative \(f'(x)\) will be \(nx^{n-1}\). This is highly efficient for polynomials and simplifies the process drastically.

Let’s consider an example: for \(f(x) = x^2\), applying the power rule gives us \(f'(x) = 2x^{2-1} = 2x\), indicating the rate of change is proportional to \(x\).

• Use the power rule when differentiating terms like \(x^3\) or \(x^{-1/2}\).
• The power rule only applies when the function follows the form \(x^n\).

In our exercise, the function is transformed into a power form that allows us to apply the power rule effectively with \(n = -1/2\).

The Chain Rule: Linking Inner and Outer Functions

The chain rule is crucial when differentiating composite functions—functions within functions. It’s used when we have an outer function \(f\) and an inner function \(g\), leading us to a composite \(f(g(x))\). It states that the derivative of \(f(g(x))\) is \(f'(g(x)) \cdot g'(x)\).

The chain rule allows us to handle these "layered" functions by differentiating the outer function and multiplying it by the derivative of the inner function.

• First, differentiate the outer function while keeping the inner function intact.
• Next, differentiate the inner function.
• Multiply both derivatives together to get the final result.

In the problem \(f(x) = (25+x^2)^{-1/2}\), the outer function is \(x^{-1/2}\), and the inner function is \(25+x^2\), via the chain rule, we coordinate these derivatives to compute \(-x(25+x^2)^{-3/2}\). This technique ensures that we properly address the dependency between the inner and outer functions.