/*! 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 5 Find \(d^{3} y / d x^{3}\). $$... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 5

Find \(d^{3} y / d x^{3}\). $$ y=\sin (7 x) $$

Short Answer

Expert verified
The third derivative is \(-343 \cos(7x)\).

Step by step solution

01

Understand the problem

We need to find the third derivative of the function \(y = \sin(7x)\) with respect to \(x\). This means we'll differentiate the function three times sequentially.
02

First Derivative

Find the first derivative of \(y\) with respect to \(x\).Given \(y = \sin(7x)\), we apply the chain rule:\[ \frac{dy}{dx} = 7 \cos(7x) \].
03

Second Derivative

Find the second derivative by differentiating the first derivative.Starting from \(\frac{dy}{dx} = 7 \cos(7x)\), apply the chain rule again:\[ \frac{d^2y}{dx^2} = -49 \sin(7x) \].
04

Third Derivative

Find the third derivative by differentiating the second derivative.Starting from \(\frac{d^2y}{dx^2} = -49 \sin(7x)\), apply the chain rule:\[ \frac{d^3y}{dx^3} = -343 \cos(7x) \].

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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 used in calculus for finding derivatives of composite functions. When you have a function within another function, like in our case where it's a trigonometric function composed with a linear function, the chain rule becomes essential.

To apply the chain rule, follow these steps:
  • Differentiate the outer function, keeping the inner function unchanged.
  • Multiply this result by the derivative of the inner function.

In our example, the original function is given by \( y = \sin(7x) \).
When differentiating \( \sin(7x) \), treat \( \sin(u) \) as the outer function and \( u=7x \) as the inner function.
The derivative of \( \sin(u) \) is \( \cos(u) \), and the derivative of \( u \), which is \( 7x \), is 7.
Thus, the first derivative becomes \( \frac{dy}{dx} = 7 \cos(7x) \).
Trigonometric Differentiation
Trigonometric differentiation involves finding derivatives of trigonometric functions. It's an essential part of calculus, especially when combined with the chain rule.

Some basic derivatives of trigonometric functions include:
  • The derivative of \( \sin(x) \) is \( \cos(x) \)
  • The derivative of \( \cos(x) \) is \( -\sin(x) \)

In the exercise, we differentiate the trigonometric function multiple times. Starting with \( \sin(7x) \), the chain rule was applied to find its derivative, giving us \( 7 \cos(7x) \).
For the second derivative, we differentiate \( \cos(7x) \), which becomes \( -\sin(7x) \), and the outer derivative multiplies again by 7, resulting in \( -49 \sin(7x) \).
Higher-Order Derivatives
Higher-order derivatives refer to the process of differentiating a function multiple times. This concept is crucial for tasks involving rates of change and concavity of functions.

The third derivative is a higher-order derivative found by differentiating a function three times sequentially.
Starting with \( y = \sin(7x) \), the third derivative involves differentiating the first and second derivatives.
  • First derivative: \( \frac{dy}{dx} = 7 \cos(7x) \)
  • Second derivative: \( \frac{d^2y}{dx^2} = -49 \sin(7x) \)
  • Third derivative: \( \frac{d^3y}{dx^3} = -343 \cos(7x) \)

The third derivative, \( \frac{d^3y}{dx^3} = -343 \cos(7x) \), is particularly useful for understanding the behavior of functions, like checking for points of inflection or analyzing motion.

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Most popular questions from this chapter

Suppose \(f\) is differentiable. If we use the approximation \(f(x+h) \approx f(x)+f^{\prime}(x) h\) the error is \(\varepsilon(h)=f(x+h)-\) \(f(x)-f^{\prime}(x) h .\) Show that(a) \(\lim _{h \rightarrow 0} \varepsilon(h)=0\) and (b) \(\lim _{h \rightarrow 0} \frac{\varepsilon(h)}{h}=0\).

Where does the tangent line to \(y=\left(x^{2}+1\right)^{-2}\) at \(\left(1, \frac{1}{4}\right)\) cross the \(x\) -axis?

If \(y=x^{4}+2 x\), find the values of \(\Delta y\) and \(d y\) in each case. (a) \(x=2\) and \(d x=\Delta x=1\) (b) \(x=2\) and \(d x=\Delta x=0.005\)

Show that the graphs of \(2 x^{2}+y^{2}=6\) and \(y^{2}=4 x\) intersect at right angles.

A wheel centered at the origin and of radius 10 centimeters is rotating counterclockwise at a rate of 4 revolutions per second. A point \(P\) on the rim is at \((10,0)\) at \(t=0\). (a) What are the coordinates of \(P\) at time \(t\) seconds? (b) At what rate is \(P\) rising (or falling) at time \(t=1\) ?
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