Problem 42 - Calculus Single Variable

Question

A function \(f\) is given. Calculate \(f^{\prime}(x)\). $$ f(x)=\sqrt{1+x^{2}} $$

Step-by-Step Solution

Answer

The derivative of the function is \( f^{ ext{'}(x)} = \frac{x}{\sqrt{1+x^2}} \).

Step 1: Understanding the Problem

We need to find the derivative of the function \( f(x) = \sqrt{1 + x^2} \). This involves using derivative rules for functions like the chain rule and power rule.

Step 2: Rewrite the Function

Rewrite \( f(x) = \sqrt{1 + x^2} \) as \( f(x) = (1 + x^2)^{1/2} \) to make it easier to use the power rule.

Step 3: Apply the Chain Rule

The chain rule states that the derivative of \( (g(x))^n \) is \( n(g(x))^{n-1} \cdot g^{ ext{'}(x)} \). For \( f(x) = (1 + x^2)^{1/2} \), we set \( g(x) = 1 + x^2 \) and \( n = 1/2 \).

Step 4: Differentiate the Outer Function

Using the power rule, differentiate the outer function: \( \frac{d}{dx}((1+x^2)^{1/2}) = \frac{1}{2}(1+x^2)^{-1/2} \cdot g^{ ext{'}(x)} \).

Step 5: Differentiate the Inner Function

Calculate \( g^{ ext{'}(x)} \) where \( g(x) = 1 + x^2 \). Here, \( g^{ ext{'}(x)} = \frac{d}{dx}(1 + x^2) = 2x \).

Step 6: Combine Results Using the Chain Rule

Combine the derivative of the outer function with the inner function's derivative. So, \( f^{ ext{'}(x)} = \frac{1}{2}(1+x^2)^{-1/2} \cdot 2x \).

Step 7: Simplify the Expression

Simplify the expression obtained. The \( 2 \) and \( \frac{1}{2} \) cancel out, leaving \( f^{ ext{'}(x)} = \frac{x}{\sqrt{1+x^2}} \).

Key Concepts

Understanding the Chain Rule

In calculus, the chain rule is an essential technique for finding the derivative of a composition of functions. This rule helps us handle situations where one function is nested inside another. Think of it like peeling an onion from the outside in.

The chain rule is formally written as:

• If you have a composite function defined as \( f(x) = (g(x))^n \), then the derivative \( f'(x) \) can be found using the formula: \( n(g(x))^{n-1} \cdot g'(x) \).

In simple terms, it tells us to:

• Differentiate the outer function while keeping the inside function unchanged temporarily.
• Then, multiply that result by the derivative of the inside function.

In the case of our exercise, the chain rule helps to find the derivative of \( f(x) = \sqrt{1 + x^2} \) by treating \( (1 + x^2)^{1/2} \) as the outer function, allowing us to focus on the inner part \( 1 + x^2 \).
Applying the chain rule requires practice, but once mastered, it becomes an incredibly powerful tool for tackling complex derivative problems.

Exploring the Power Rule

The power rule is one of the simplest and most frequently used rules in differential calculus. It provides an easy way to differentiate functions of the form \( x^n \), where \( n \) is any real number.

According to the power rule:

• The derivative of \( x^n \) is \( nx^{n-1} \). This means you bring down the exponent as a coefficient and then reduce the exponent by one.

This rule applies seamlessly in our problem when addressing the outer function, which is expressed as \( (1 + x^2)^{1/2} \).
By applying the power rule:

• The derivative of \( (1 + x^2)^{1/2} \) will be \( \frac{1}{2}(1 + x^2)^{-1/2} \).

Combining it with the chain rule (to handle the nested \( 1 + x^2 \)), helps us solve more advanced calculus problems efficiently.
Being comfortable with the power rule is crucial, as it serves as a foundation for more complex differentiation tasks in calculus.

Connecting with Calculus

At the heart of calculus lies the concept of limits, derivatives, and integrals, which help us understand the behavior of functions, especially those that describe change and motion.

Calculus is split into two main branches:

• Differential calculus, which deals with rates of change and slopes of curves.
• Integral calculus, which focuses on areas under curves and accumulation of quantities.

In our specific exercise, we're diving into differential calculus to compute the derivative of a function—\( f(x) = \sqrt{1 + x^2} \)—a task made possible with techniques like the chain rule and power rule.
Understanding calculus is essential for multiple fields, including physics, engineering, and economics, as it provides the tools needed to model systems and predict future behavior.
Remember, the goal of calculus is to make sense of quantities in a continuous system and how they interact or change over time or space.