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Chapter 3: Problem 25

In Exercises \(1-56,\) find the derivatives. Assume that \(a\) and \(b\) are constants. $$y=\sqrt{s^{3}+1}$$

Short Answer

Expert verified
The derivative is \( \frac{3s^2}{2\sqrt{s^3 + 1}} \).

Step by step solution

01

Identify the function

The function given is \( y = \sqrt{s^3 + 1} \). This is an expression involving a square root, which can be rewritten for easier differentiation.
02

Rewrite the function for differentiation

Rewrite the square root as a power: \( y = (s^3 + 1)^{1/2} \). This representation will help us apply the chain rule when finding the derivative.
03

Apply the chain rule

The chain rule states that the derivative of \( y = (u(s))^n \) is \( \frac{dy}{ds} = nu^{n-1}(s) \cdot \frac{du}{ds} \). Here, \( u(s) = s^3 + 1 \) and \( n = 1/2 \).
04

Differentiate using the chain rule

Apply the power rule and the derivative of \( u(s) = s^3 + 1 \). We have: \[ \frac{dy}{ds} = \frac{1}{2}(s^3 + 1)^{-1/2} \cdot 3s^2 \].
05

Simplify the derivative

Simplify the expression: \[ \frac{dy}{ds} = \frac{3s^2}{2\sqrt{s^3 + 1}} \]. This is the derivative of the original function.

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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 used for finding the derivative of composite functions. A composite function is one where a function is nested inside another. If we break it down:
  • If you have a function of the form \( f(g(x)) \), the chain rule helps you differentiate it by focusing separately on the outer function \( f \) and the inner function \( g \).
  • The derivative using the chain rule is \( f'(g(x)) \times g'(x) \).
In the solution for this exercise, the chain rule is used to differentiate the function \( y = (s^3 + 1)^{1/2} \). Here, the outer function is \( (u)^{1/2} \) and the inner function is \( u = s^3 + 1 \). First, you differentiate the outer function while keeping \( u \) as is, and then multiply by the derivative of \( u \). This systematic approach simplifies working with complex expressions.
Power Rule
The power rule is a straightforward and essential tool in differentiation, especially when dealing with polynomial expressions. The rule states:
  • If \( f(x) = x^n \), then the derivative \( f'(x) = nx^{n-1} \).
In this exercise, it is necessary to use the power rule after applying the chain rule. The term \( (s^3 + 1)^{1/2} \) identifies the 'power' we're dealing with. Therefore:
  • The exponent \( 1/2 \) is brought down in front as a coefficient.
  • The power of the expression is then decreased by 1, resulting in \( (s^3 + 1)^{-1/2} \).
This application of the power rule makes handling the derivative of non-linear functions much easier.
Differentiation
Differentiation is the process of finding the derivative, which is essentially the rate of change of a function. This process is the backbone of calculus problem-solving and enables us to understand how functions behave. When performing differentiation:
  • Identify the type of function (e.g., polynomial, exponential, trigonometric).
  • Select appropriate differentiation rules (like the chain or power rules) to apply.
  • Simplify the result to make it more interpretable.
In this exercise, differentiation helps find how the function \( y = \sqrt{s^3 + 1} \) changes as \( s \) changes. By breaking down the function using the chain and power rules, you obtain the derivative \( \frac{3s^2}{2\sqrt{s^3 + 1}} \), representing the rate of change of \( y \) with respect to \( s \).
Calculus Problem Solving
Calculus problem-solving often involves applying systematic strategies to find derivatives and integrals. For instance, when tasked with finding the derivative of \( y = \sqrt{s^3 + 1} \), follow these steps:
  • Understand the function format: Recognize that it can be rewritten for easier manipulations.
  • Apply appropriate techniques: Use the chain rule for composite functions and the power rule for functions with exponents.
  • Perform calculations carefully: Differentiate step-by-step and make sure each transformation is accurate.
  • Simplify the result: Ensure the final answer is as clear as possible, such as \( \frac{3s^2}{2\sqrt{s^3 + 1}} \).
By practicing these methods, developing proficiency in calculus becomes attainable. The systematic use of rules and careful analysis turns complex functions into a series of solvable steps.

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Most popular questions from this chapter

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