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Chapter 2: Problem 13

Differentiate the function. \(A(s)=-\frac{12}{s^{5}}\)

Short Answer

Expert verified
The derivative of the function is \( A'(s) = \frac{60}{s^6} \).

Step by step solution

01

Rewrite the function

First, rewrite the given function in a form that makes it easier to differentiate. The given function is \[ A(s) = -\frac{12}{s^5} \] This can be rewritten using a power of a negative exponent for easier differentiation:\[ A(s) = -12s^{-5} \]
02

Apply the power rule of differentiation

Differentiate the function using the power rule, which states that if \[ f(s) = as^n \] then \[ f'(s) = nas^{n-1} \].For \[ A(s) = -12s^{-5} \], \[-12 \] is the constant, and \[-5 \] is the exponent. The derivative is:\[ A'(s) = -12(-5)s^{-5-1} \]\[ A'(s) = 60s^{-6} \]
03

Simplify the derivative

Simplify the expression obtained from the differentiation in the previous step. Rewrite the derivative in terms of positive exponents if needed:\[ A'(s) = 60s^{-6} \]This can also be expressed as a fraction:\[ A'(s) = \frac{60}{s^6} \]

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Key Concepts

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

Power Rule
Differentiating functions often involves using the power rule, a fundamental concept in calculus. The power rule is a quick way to find the derivative of a polynomial function. It states: if you have a function of the form \(f(x) = ax^n\), its derivative is \(f'(x) = nax^{n-1}\). This powerful tool simplifies the process of differentiation by establishing a direct formula.
  • Identify the coefficient \(a\) and the exponent \(n\).
  • Multiply the exponent \(n\) by the coefficient \(a\).
  • Subtract one from the exponent to find the new exponent for \(x\).
By following these steps, you can quickly differentiate polynomial functions. The power rule is applicable to any real number exponent, even when it is negative or a fraction.
Negative Exponents
Negative exponents offer a compact way of expressing functions, especially for differentiation. When you encounter a negative exponent, it's important to understand its implications:
  • A negative exponent \(x^{-n}\) equals \(\frac{1}{x^n}\).
  • This transformation allows expressions to be rewritten, facilitating differentiation using standard rules like the power rule.
Consider this example: turning \(-\frac{12}{s^5}\) into \(-12s^{-5}\). This rewrite is crucial for differentiation, making the function much more manageable. Understanding how to manipulate and interpret negative exponents is key for solving calculus problems efficiently.
Derivatives
Derivatives represent the foundation of calculus, essentially measuring how a function changes as its input changes. When you differentiate a function, you're finding its derivative, providing insights into its rate of change. This is essential in various fields such as physics, engineering, and economics.To find a derivative:
  • Start with rewriting the function if necessary (e.g., using negative exponents).
  • Apply rules like the power rule to calculate derivative values.
  • Simplify the resulting expression to its simplest form.
The derivative of a function, often denoted as \(f'(x)\) or \(\frac{df}{dx}\), reveals critical information such as slope, velocity, and acceleration, making it a versatile tool in analysis and interpretation.
Function Rewriting
Function rewriting is a technique that transforms expressions into a more usable format, particularly for differentiation and integration. It involves adjusting the form of a function to simplify the application of calculus rules.
  • Simplifies complex exponents or terms into manageable pieces.
  • Makes application of rules like the power rule straightforward.
  • Enhances clarity by standardizing expression forms.
For example, in the function \(-\frac{12}{s^5}\), rewriting it as \(-12s^{-5}\) turns a division problem into a multiplication task with a clearer application of the power rule. By mastering function rewriting, tackling calculus problems becomes more systematic and less prone to error.

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Most popular questions from this chapter

Suppose that we don't have a formula for \(g(x)\) but we know that \(g(2)=-4\) and \(g^{\prime}(x)=\sqrt{x^{2}+5}\) for all \(x .\) (a) Use a linear approximation to estimate \(g(1.95)\) and \(g(2.05) .\) (b) Are your estimates in part (a) too large or too small? Explain.

Brain weight \(B\) as a function of body weight \(W\) in fish has been modeled by the power function \(B=0.007 W^{2 / 3}\) where \(B\) and \(W\) are measured in grams. A model for body weight as a function of body length \(L\) (measured in centimeters) is \(W=0.12 L^{2.53} .\) If, over 10 million years, the average length of a certain species of fish evolved from 15 \(\mathrm{cm}\) to 20 \(\mathrm{cm}\) at a constant rate, how fast was this species' brain growing when the average length was 18 \(\mathrm{cm} ?\)

$$ \begin{array}{l}{\text { If } F(x)=f(3 f(4 f(x))), \text { where } f(0)=0 \text { and } f^{\prime}(0)=2} \\ {\text { find } F^{\prime}(0)}\end{array} $$

The equation \(y^{\prime \prime}+y^{\prime}-2 y=\sin x\) is called a differential equation because it involves an unknown function \(y\) and its derivatives \(y^{\prime}\) and \(y^{\prime \prime}\) . Find constants \(A\) and \(B\) such that the function \(y=A \sin x+B \cos x\) satisfies this equation. (Dif- ferential equations will be studied in detail in Section 7.7 )

A trough is 10 \(\mathrm{ft}\) long and its ends have the shape of isosceles triangles that are 3 ft across at the top and have a height of 1 \(\mathrm{ft}\) . If the trough is being filled with water at a rate of 12 \(\mathrm{ft}^{3} / \mathrm{min}\) , how fast is the water level rising when the water is 6 inches deep?
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