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Chapter 1: Problem 8

Find the first derivatives. Find \(\frac{d}{d s} \sqrt{s^{2}+1}\).

Short Answer

Expert verified
\(\frac{s}{\sqrt{s^2 + 1}}\)

Step by step solution

01

Identify the Function

The function given is \(f(s) = \sqrt{s^2 + 1}\).
02

Apply the Chain Rule

The chain rule states that \(\frac{d}{ds}[f(g(s))] = f'(g(s)) \cdot g'(s)\). Here, \(f(u) = \sqrt{u}\) and \(g(s) = s^2 + 1\).
03

Differentiate the Outer Function

Differentiate \f(u) = \sqrt{u}\ with respect to \(u\): \[\frac{d}{du}[\sqrt{u}] = \frac{1}{2\sqrt{u}}\].
04

Differentiate the Inner Function

Differentiate the inner function \(g(s) = s^2 + 1\) with respect to \s\: \[\frac{d}{ds}[s^2 + 1] = 2s\].
05

Combine the Derivatives

Using the chain rule, combine the derivatives: \[\frac{d}{ds}[\sqrt{s^2 + 1}] = \frac{1}{2\sqrt{s^2 + 1}} \cdot 2s = \frac{s}{\sqrt{s^2 + 1}}\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

chain rule
The chain rule is a fundamental technique in calculus for finding the derivative of a composite function. It connects the rate of change of the outer function to the rate of change of the inner function. In simple terms, if you have a function inside another function, you need to differentiate both functions and then multiply the results.

For example, if you have a function like \(h(x) = f(g(x))\), the chain rule helps you find \(\frac{dh}{dx}\) by this formula: \(\frac{dh}{dx} = f'(g(x)) \times g'(x) \).

In our exercise, we have \(f(s) = \sqrt{s^2 + 1}\). We treat \(s^2 + 1\) as the inner function and \(\sqrt{}\)\
differentiation
Differentiation is the process of finding the derivative of a function. The derivative gives the rate at which a function is changing at any given point. Think of it as finding the slope of the function at a specific point.

In our problem, we need to find the derivative of \(\sqrt{s^2 + 1}\)\
outer function
The outer function is the function that is applied last in a composition of functions. For example, in the function \(f(g(x))\), \f(x)\ is the outer function.

In our exercise, the outer function is \(\sqrt{u}\)\
inner function
The inner function is the function that is applied first in a composition of functions. For example, in the function \(f(g(x))\), \g(x)\ is the inner function.

In our exercise, the inner function is \(s^2 + 1\). We need to differentiate this first before using the chain rule.

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Most popular questions from this chapter

Use limits to compute the following derivatives. \(f^{\prime}(0)\), where \(f(x)=x^{2}+2 x+2\)

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