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Chapter 3: Problem 26

In Exercises \(19-30,\) find \(d y\) $$ y=\cos \left(x^{2}\right) $$

Short Answer

Expert verified
The differential is \( dy = -2x \sin(x^2) \, dx \).

Step by step solution

01

Identify the Function and Derivative Rule

The given function is \( y = \cos(x^2) \). Here, we need to find \( dy \), the differential of \( y \). The function has an inner function \( u = x^2 \) and an outer function \( \cos(u) \). To differentiate this, we'll use the chain rule.
02

Differentiate the Inner Function

Differentiate the inner function \( u = x^2 \) with respect to \( x \). The derivative is \( \frac{du}{dx} = 2x \). This gives us the rate of change of the inner function.
03

Differentiate the Outer Function

Differentiate the outer function \( y = \cos(u) \) with respect to \( u \). The derivative is \( \frac{dy}{du} = -\sin(u) \). This tells us how the outer function changes with respect to its input.
04

Apply the Chain Rule

By the chain rule, the derivative \( \frac{dy}{dx} \) is the product of \( \frac{dy}{du} \) and \( \frac{du}{dx} \). Therefore, \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = (-\sin(u)) \cdot (2x) = -2x \sin(x^2) \).
05

Write the Differential dy

The differential \( dy \) is given by \( dy = \frac{dy}{dx} \cdot dx = -2x \sin(x^2) \cdot dx \). This describes the small change in \( y \) in terms of a small change in \( x \).

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Key Concepts

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

Understanding Derivatives
A derivative provides a measure of how a function changes as its input changes. It's essentially the rate of change or the slope of the function at any given point. In calculus, we denote derivatives by symbols such as \( dy/dx \) or \( f'(x) \). These symbols represent the change in the dependent variable \( y \) with respect to the change in the independent variable \( x \).
  • Calculating derivatives is foundational in differential calculus because it allows us to understand how functions behave and change over time or space.
  • The process involves applying various rules of differentiation to find these rates of change.
In our exercise, we are dealing with a composite function, \( y = \cos(x^2) \), and we must apply a specific technique called the chain rule to find its derivative.
The Chain Rule Explained
The chain rule is a fundamental technique in differential calculus for finding the derivative of a composite function. When you have a function nested inside another function, or in other words, "a function of a function," you must use the chain rule.
Here's how the chain rule works:
  • Identify the outer function and the inner function. For example, in our exercise, the inner function is \( u = x^2 \) and the outer function is \( \, y = \cos(u) \).
  • First, differentiate the outer function with respect to the inner function (\( u \)).
  • Then, differentiate the inner function with respect to \( x \).
  • Finally, multiply these derivatives together to get the derivative of the composite function \( y \) with respect to \( x \).
Thus, using the chain rule, we've derived \( dy/dx = -2x \sin(x^2) \). This shows how the function \( y = \cos(x^2) \) changes concerning \( x \).
Trigonometric Functions in Calculus
Trigonometric functions like sine, cosine, and tangent are part of essential mathematical operations and appear frequently in calculus. Differentiating trigonometric functions follows specific rules:
  • The derivative of \( \, \sin(x) \) is \( \cos(x) \).
  • The derivative of \( \cos(x) \) is \(-\sin(x) \).
  • The derivative of \( \tan(x) \) is \( \sec^2(x) \).
In the given function \( y = \cos(x^2) \), knowing that the derivative of \( \cos(u) \) is \(-\sin(u)\) is crucial to solving the problem. Applying these rules helps determine the rate of change for trigonometric functions, portraying how they oscillate or vary.
Differentiation Techniques
Being able to differentiate various types of functions is crucial in calculus. Different functions require different techniques:
  • Power Rule: Used for differentiating functions like \( x^n \), where the derivative is \( nx^{n-1} \).
  • Product Rule: Used when differentiating the product of two functions.
  • Quotient Rule: Utilized for functions that are one divided by another.
  • Chain Rule: As covered, ideal for composite functions.
In our example, we applied the power rule for the inner function \( x^2 \) and the chain rule for the whole expression \( \cos(x^2) \). Being familiar with these techniques allows for accurate calculation of derivatives, assisting in solving more complex calculus problems.

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Most popular questions from this chapter

Falling meteorite The velocity of a heavy meteorite entering Earth's atmosphere is inversely proportional to \(\sqrt{s}\) when it is \(s\) \(\mathrm{km}\) from Earth's center. Show that the meteorite's acceleration is inversely proportional to \(s^{2} .\)

Particle acceleration A particle moves along the \(x\) -axis with velocity \(d x / d t=f(x)\) . Show that the particle's acceleration is \(f(x) f^{\prime}(x) .\)

In Exercises \(67-70\) , use a CAS to estimate the magnitude of the error in using the linearization in place of the function over a specified interval \(I\) . Perform the following steps: a. Plot the function \(f\) over \(I .\) b. Find the linearization \(L\) of the function at the point \(a\) . c. Plot \(f\) and \(L\) together on a single graph. d. Plot the absolute error \(|f(x)-L(x)|\) over \(I\) and find its maximum value. e. From your graph in part (d), estimate as large a \(\delta > 0\) as you can, satisfying $$ |x-a|<\delta \quad \Rightarrow \quad|f(x)-L(x)|<\epsilon $$ for \(\epsilon=0.5,0.1,\) and 0.01 . Then check graphically to see if your \(\delta\) -estimate holds true. $$ f(x)=x^{3}+x^{2}-2 x, \quad[-1,2], \quad a=1 $$

In Exercises \(67-70\) , use a CAS to estimate the magnitude of the error in using the linearization in place of the function over a specified interval \(I\) . Perform the following steps: a. Plot the function \(f\) over \(I .\) b. Find the linearization \(L\) of the function at the point \(a\) . c. Plot \(f\) and \(L\) together on a single graph. d. Plot the absolute error \(|f(x)-L(x)|\) over \(I\) and find its maximum value. e. From your graph in part (d), estimate as large a \(\delta > 0\) as you can, satisfying $$ |x-a|<\delta \quad \Rightarrow \quad|f(x)-L(x)|<\epsilon $$ for \(\epsilon=0.5,0.1,\) and 0.01 . Then check graphically to see if your \(\delta\) -estimate holds true. $$ f(x)=\sqrt{x}-\sin x, \quad[0,2 \pi], \quad a=2 $$

In Exercises \(19-30,\) find \(d y\) $$ y=3 \csc (1-2 \sqrt{x}) $$
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