/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 9 Find \(\frac{d y}{d x}\). $$y=... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 9

Find \(\frac{d y}{d x}\). $$y=x^{-6}$$

Short Answer

Expert verified
\(\frac{d y}{d x} = -6 x^{-7}\).

Step by step solution

01

Identify the function to differentiate

The given function to differentiate is \(y = x^{-6}\).
02

Apply the power rule of differentiation

To differentiate \(y = x^{-6}\), use the power rule of differentiation, which states \(\frac{d}{dx}(x^n) = nx^{n-1}\).
03

Differentiate the function

Apply the power rule to the given function: \[\frac{d y}{d x} = \frac{d}{d x} (x^{-6}) = -6x^{-7}.\]
04

Simplify the result

The simplified form of the derivative is \[\frac{d y}{d x} = -6x^{-7}.\]

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Key Concepts

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

derivative of a function
The derivative of a function tells us how the function value changes as its input changes. If we have a function y = x^n, the derivative helps us find the rate at which y changes with respect to x. In simple terms, it shows us the slope of the function at any point.
In this exercise, we are looking at a function y = x^{-6}. Our goal is to find how y changes as x changes.
differentiation techniques
Differentiation is a key concept in calculus that involves finding the derivative of a function. The process may sound complex, but there are established techniques that simplify it:
  • Power Rule: This is one of the most commonly used techniques. If you have a function in the form of x^n, the power rule states that the derivative is given by nx^{n-1}.
  • Product Rule: This rule is used when differentiating a product of two functions.
  • Quotient Rule: This is useful for differentiating a ratio of two functions.
  • Chain Rule: This technique helps differentiate composite functions.
In the given exercise, we only need the power rule. As per the step-by-step solution, we have y = x^{-6}. Applying the power rule, we get the derivative as(nx^{n-1}). Hence, the derivative of x^{-6} is -6x^{-7}.
calculus
Calculus is a branch of mathematics focusing on change and motion. It consists of two main parts: differentiation and integration.
  • Differentiation: This deals with finding the derivative, giving us the rate of change of a function.
  • Integration: While differentiation looks at the rate of change, integration helps find the area under curves and the total accumulation of quantities.
In the context of our exercise, we're focusing on differentiation. Specifically, the power rule simplifies our task. By understanding each step clearly, you're setting a strong foundation for more advanced concepts in calculus. Keep practicing these techniques to become proficient.

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Most popular questions from this chapter

Tongue-Tied Sauces, Inc., finds that the revenue, in dollars, from the sale of \(x\) bottles of barbecue sauce is given by \(R(x)=7.5 x^{0.7} .\) Find the rate at which average revenue is changing when 81 bottles of barbecue sauce have been produced.

For each function, find the interval(s) for which \(f^{\prime}(x)\) is positive. $$f(x)=\frac{1}{3} x^{3}-x^{2}-3 x+5$$

Differentiate. $$y=\left(\frac{x}{\sqrt{x-1}}\right)^{3}$$

The function \(f(x)=x^{3}-x^{2}\) (mentioned after Example 8 ) appears to be always increasing, or possibly flat, on the default viewing window of the TI-83. a) Graph the function in the default window; then zoom in until you see a small interval in which \(f\) is decreasing. b) Use the derivative to determine the point(s) at which the graph has horizontal tangent lines. c) Use your result from part (b) to infer the interval for which \(f\) is decreasing. Does this agree with your calculator's image of the graph? d) Is it possible there are other intervals for which \(f\) is decreasing? Explain why or why not.

Utility is a type of function that occurs in economics. When a consumer receives \(x\) units of a product, a certain amount of pleasure, or utility, \(U\), is derived. Suppose that the utility related to the number of tickets \(x\) for a ride at a county fair is $$U(x)=80 \sqrt{\frac{2 x+1}{3 x+4}}$$ Find the rate at which the utility changes with respect to the number of tickets bought.
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