/*! 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 14 Find \(\frac{d y}{d x}\) \(y=x... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 14

Find \(\frac{d y}{d x}\) \(y=x^{4}-7 x\)

Short Answer

Expert verified
The derivative is \( \frac{dy}{dx} = 4x^3 - 7 \).

Step by step solution

01

Identify the function

Observe that the given function is a polynomial function: \( y = x^4 - 7x \)
02

Apply the power rule to the first term

For the term \( x^4 \), the power rule states that the derivative of \( x^n \) is \( nx^{n-1} \). Applying this rule, the derivative of \( x^4 \) is \( 4x^3 \).
03

Apply the power rule to the second term

For the term \( -7x \), the power rule states that the derivative of \( x \) is \( 1 \). Therefore, the derivative of \( -7x \) is \( -7 \).
04

Combine the results

Adding the derivatives of each term together, the derivative of the function \( y = x^4 - 7x \) is \( 4x^3 - 7 \).

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Key Concepts

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

power rule
The power rule is a fundamental tool in differentiation. It simplifies finding the derivatives of polynomial terms. The rule states that if you have a term of the form \(x^n\), its derivative is \(nx^{n-1}\). This means you multiply by the exponent and then reduce the exponent by one.
For example, for \(x^4\), the power rule tells us its derivative is \(4x^3\). This is achieved by multiplying the exponent 4 by the term and decreasing its power by 1.
Understanding this rule is essential for finding the derivatives of polynomial functions efficiently. This simple rule can be applied to any term where the variable x is raised to a power.
differentiation
Differentiation is the process of computing a derivative. A derivative measures how a function's output changes as its input changes. Essentially, it answers the question: how does one variable affect another?
In the exercise, differentiation is applied to each term of the polynomial \(y = x^4 - 7x\). By differentiating each term, we find how the entire function changes.
For \(x^4\), applying differentiation using the power rule we get \(4x^3\). For \(-7x\), since the derivative of \(x\) is 1, we have \(-7\). By combining these results, we see that the derivative of the whole function is \(4x^3 - 7\).
polynomial function
A polynomial function consists of terms that are non-negative integer powers of a variable. Examples include \(x^2\), \(x^3\), and \(x^4\).These functions can be very simple or quite complex but share a common trait: their derivatives can be found using the power rule.
In the exercise, we are given the polynomial \(y = x^4 - 7x\). Each term in this function is a simple polynomial term. By applying differentiation to each term, we find their individual derivatives and combine them.
This makes polynomial functions relatively straightforward to work with in calculus, as long as you understand the power rule and differentiation.

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Most popular questions from this chapter

If \(f(x)\) is a function, then \((f \circ f)(x)=f(f(x))\) is the composition of \(f\) with itself. This is called an iterated function, and the composition can be repeated many times. For example, \((f \circ f \circ f)(x)=f(f(f(x))) .\) Iterated functions are very useful in many areas, including finance (compound interest is \(a\) simple case) and the sciences (in weather forecasting, for example). For the each function, use the Chain Rule to find the derivative.. If \(f(x)=\sqrt[3]{x},\) find \(\frac{d}{d x}[(f \circ f \circ f)(x)]\).

If \(f(x)\) is a function, then \((f \circ f)(x)=f(f(x))\) is the composition of \(f\) with itself. This is called an iterated function, and the composition can be repeated many times. For example, \((f \circ f \circ f)(x)=f(f(f(x))) .\) Iterated functions are very useful in many areas, including finance (compound interest is \(a\) simple case) and the sciences (in weather forecasting, for example). For the each function, use the Chain Rule to find the derivative. If \(f(x)=x^{2}+1,\) find \(\frac{d}{d x}[(f \circ f)(x)]\).

Find the derivative of each of the following functions analytically. Then use a calculator to check the results. $$f(x)=x \sqrt{4-x^{2}}$$

Consumer demand. Suppose that the demand function for a product is given by \(D(p)=\frac{80,000}{p}\) and that price \(p\) is a function of time given by \(p=1.6 t+9,\) where \(t\) is in days. a) Find the demand as a function of time \(t\) b) Find the rate of change of the quantity demanded when \(t=100\) days.

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