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Chapter 3: Problem 27

Find \(d y\) $$y=4 \tan \left(x^{3} / 3\right)$$

Short Answer

Expert verified
\( dy = 4 x^2 \sec^2\left(\frac{x^3}{3}\right) dx \)

Step by step solution

01

Differentiate the Outer Function

The given function is in a composite form, where the outer function is tangent, and the inner function is \( x^3/3\). Start by differentiating the outer function. The derivative of \( \tan(u)\) with respect to \( u\) is \( \sec^2(u)\). Thus, differentiate and multiply by the inner part:\[\frac{d}{du}\left(4 \tan(u)\right) = 4 \sec^2(u) \cdot \frac{du}{dx}\]
02

Differentiate the Inner Function

Next, differentiate the inner function, which is \( x^3/3\), with respect to \( x\). Applying the power rule, the derivative of \( x^3 \) is \( 3x^2\). Divide this by 3 to get:\[\frac{d}{dx}\left(\frac{x^3}{3}\right) = x^2\]
03

Apply the Chain Rule

Now use the chain rule to find \( \frac{dy}{dx} \) by combining the results from Step 1 and Step 2. Multiply the derivative of the outer function by the derivative of the inner function:\[\frac{dy}{dx} = 4 \sec^2\left(\frac{x^3}{3}\right) \cdot x^2\]
04

Write the Final Expression for dy

To express \( dy \), multiply \( \frac{dy}{dx} \) by \( dx \) as follows:\[dy = 4 x^2 \sec^2\left(\frac{x^3}{3}\right) \cdot dx\]

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Key Concepts

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

Chain Rule in Derivative Calculus
In calculus, the chain rule is a fundamental technique used to find the derivative of a composite function. A composite function is one where one function is "inside" another.
For instance, in our example the function is in the form of \( y = 4 \tan\left(\frac{x^3}{3}\right) \). Here, \( \tan \) is the outer function, and \( \frac{x^3}{3} \) is the inner function.
  • The derivative of the outer function \( \tan(u) \) with respect to \( u \) is \( \sec^2(u) \).
  • The derivative of the inner function \( \frac{x^3}{3} \) with respect to \( x \) is \( x^2 \).
By using the chain rule, we multiply the derivative of the outer function by the derivative of the inner function.
This gives us the overall derivative with respect to \( x \). It simplifies finding derivatives of more complex functions by breaking them into manageable parts.
Differentiation Basics
Differentiation is an essential concept in calculus, focusing on finding the rate at which a function changes. It's like finding how fast something is happening, such as speed.
When differentiating a function, you're essentially finding its slope or its rate of change at any given point.
In this exercise, the function \( x^3/3 \) was differentiated using the power rule.
  • The power rule states: for \( x^n \), its derivative is \( nx^{n-1} \).
  • For \( x^3 \), applying the power rule gives \( 3x^2 \).
  • Dividing by 3, since our inner function is \( x^3/3 \), results in \( x^2 \).
Differentiating allows you to explore how a function behaves, providing insights into curves and their tangents. This step helps ensure all parts of a composite function are correctly accounted for.
Understanding Composite Functions
Composite functions are those created by combining two or more functions. They're represented as \((f \circ g)(x)\) which means \( f(g(x)) \).
This simply implies that you're applying one function to the result of another.
  • In the given exercise, \( \tan\left(\frac{x^3}{3}\right) \) is a composite function.
  • Here, \( \tan \) is applied to the function \( \frac{x^3}{3} \).
When dealing with composite functions, calculating derivatives involves peeling back the layers, similar to an onion, to differentiate each part using techniques like the chain rule.
Composite functions help model real-world processes where one change affects another, making them crucial in both mathematics and practical applications.

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Most popular questions from this chapter

Find the derivative of \(y\) with respect to the given independent variable. $$y=\log _{25} e^{x}-\log _{5} \sqrt{x}$$

The effect of flight maneuvers on the heart The amount of work done by the heart's main pumping chamber, the left ventricle, is given by the equation $$ W=P V+\frac{V \delta v^{2}}{2 g} $$ where \(W\) is the work per unit time, \(P\) is the average blood pressure, \(V\) is the volume of blood pumped out during the unit of time, \(\delta\) ("delta") is the weight density of the blood, \(v\) is the average velocity of the exiting blood, and \(g\) is the acceleration of gravity. When \(P, V, \delta,\) and \(v\) remain constant, \(W\) becomes a function of \(g\), and the equation takes the simplified form $$ W=a+\frac{b}{g}(a, b \text { constant }) $$ As a member of NASA's medical team, you want to know how sensitive \(W\) is to apparent changes in \(g\) caused by flight maneuvers, and this depends on the initial value of \(g\). As part of your investigation, you decide to compare the effect on \(W\) of a given change \(d g\) on the moon, where \(g=5.2 \mathrm{ft} / \mathrm{sec}^{2}\), with the effect the same change \(d_{8}\) would have on Earth, where \(g=32 \mathrm{ft} / \mathrm{sec}^{2}\). Use the simplified equation above to find the ratio of \(d W_{\text {moon }}\) to \(d W_{\text {Barth }}\)

Graph the rational function \(y=\left(2-x^{2}\right) / x^{2} .\) Then graph \(y=\) \(\cos \left(2 \sec ^{-1} x\right)\) in the same graphing window. What do you see? Explain.

a. Let \(Q(x)=b_{0}+b_{1}(x-a)+b_{2}(x-a)^{2}\) be a quadratic approximation to \(f(x)\) at \(x=a\) with these properties: i. \(Q(a)=f(a)\) ii. \(Q^{\prime}(a)=f^{\prime}(a)\) iii. \(Q^{\gamma}(a)=f^{\prime \prime}(a)\) Determine the coefficients \(b_{0}, b_{1},\) and \(b_{2}\) b. Find the quadratic approximation to \(f(x)=1 /(1-x)\) at \(x=0\) c. Graph \(f(x)=1 /(1-x)\) and its quadratic approximation at \(x=0 .\) Then zoom in on the two graphs at the point (0,1) Comment on what you see. d. Find the quadratic approximation to \(g(x)=1 / x\) at \(x=1\) Graph \(g\) and its quadratic approximation together. Comment on what you see. e. Find the quadratic approximation to \(h(x)=\sqrt{1+x}\) at \(x=0 .\) Graph \(h\) and its quadratic approximation together. Comment on what you see. I. What are the linearizations of \(f, g,\) and \(h\) at the respective points in parts (b), (d), and (e)?

Find the derivative of \(y\) with respect to the given independent variable. $$y=\log _{2}\left(\frac{x^{2} e^{2}}{2 \sqrt{x+1}}\right)$$
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