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Chapter 3: Problem 21

$$\text { Evaluate the derivatives of the following functions.}$$ $$f(u)=\csc ^{-1}(2 u+1)$$

Short Answer

Expert verified
Question: Find the derivative of the function \(f(u)=\csc ^{-1}(2 u+1)\). Answer: The derivative of the function is \(\frac{df}{du} = -\frac{2}{|(2u+1)|\sqrt{(2u+1)^2-1}}\).

Step by step solution

01

Identify the inner function

The given function is \(f(u) = \csc ^{-1}(2 u + 1)\), and the inner function is \(g(u) = 2 u + 1\).
02

Find the derivative of the inner function

Now let's find the derivative of g(u) with respect to u: $$\frac{d}{du}(2u+1) = 2$$
03

Apply the chain rule

To find the derivative of \(f(u)\) with respect to \(u\), we will apply the chain rule: $$\frac{df}{du} = \frac{d}{du}(\csc^{-1}(g(u))) \times \frac{dg}{du}$$
04

Substitute the derivatives and the inner function

Now we substitute our earlier results for \(g(u)\) and its derivative, as well as the formula for the derivative of the inverse cosecant: $$\frac{df}{du} = -\frac{1}{|g(u)|\sqrt{g(u)^2-1}} \times 2$$
05

Substitute the expression for g(u)

Finally, we substitute the expression for \(g(u) = 2u + 1\): $$\frac{df}{du} = -\frac{2}{|(2u+1)|\sqrt{(2u+1)^2-1}}$$ That's the derivative of the given function.

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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 concept in calculus used to find the derivative of composite functions. Composite functions are functions that have another function inside them. Think of it like a Russian nesting doll, where you have layers of functions.
  • Suppose you have a function \( y = f(g(x)) \), where \( f \) and \( g \) are functions of \( x \).
  • The Chain Rule states that the derivative of \( y \) with respect to \( x \) is the product of the derivative of \( f \) with respect to \( g \) and the derivative of \( g \) with respect to \( x \): \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).

In our exercise, applying the Chain Rule means differentiating \( \csc^{-1}(2u+1) \). We first found the derivative of the inner function \( 2u+1 \). This is just 2.
Then, we need the derivative of \( \csc^{-1}(x) \) with respect to \( x \), which is used later when combining with the derivatives we found. Consider the Chain Rule as linking these parts together to get our result.
Inverse Cosecant Function
The inverse cosecant function, represented as \( \csc^{-1}(x) \), is the inverse function of \( \csc(x) \). Understanding this function is key when you derive its counterpart.
  • First, recognize that \( y = \csc^{-1}(x) \) implies that \( \csc(y) = x \).
  • The domains for the inverse trig functions like \( \csc^{-1} \) are limited, typically \( y \in [-\frac{\pi}{2}, 0) \cup (0, \frac{\pi}{2}] \), because these functions have restrictions where they are defined.
  • The derivative of \( \csc^{-1}(x) \) is \( -\frac{1}{|x|\sqrt{x^2-1}} \), crucial for using in derivatives involving inverse trigonometric functions.

In the given exercise, it's important to recall this derivative formula since we substitute \( x = (2u+1) \).
This step determines how changes in \( 2u+1 \) reflect through to changes in the inverse cosecant, thanks to this formula.
Derivative Calculation
In calculating the derivative of \( f(u) = \csc^{-1}(2u+1) \), we bring together understanding of the chain rule and the function \( \csc^{-1} \).Let's break it down:
  • First, identify the inner function: \( g(u) = 2u + 1 \).
  • The derivative of this function, \( g'(u) \), is 2, as calculated straightforwardly from the power rule.
  • Next, utilize the formula for the derivative of the inverse cosecant, \( -\frac{1}{|x|\sqrt{x^2-1}} \), where \( x \) becomes \( g(u) \).

Applying the Chain Rule:
  • Substitute \( g(u) = 2u+1 \) and \( g'(u) = 2 \) into the formula from the chain rule: \( \frac{df}{du} = \frac{d}{du}(\csc^{-1}(g(u))) \cdot g'(u) \).
  • This gives us \( \frac{df}{du} = -\frac{2}{|(2u+1)|\sqrt{(2u+1)^2-1}} \), due to the multiplication and substitution processes previously learned.

The result is your complete derivative, derived by connecting all parts like puzzle pieces to see how each affects the outcome. Remembering these relationships helps in handling similar problems with ease.

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Most popular questions from this chapter

Calculate the following derivatives using the Product Rule. $$\begin{array}{lll} \text { a. } \frac{d}{d x}\left(\sin ^{2} x\right) & \text { b. } \frac{d}{d x}\left(\sin ^{3} x\right) & \text { c. } \frac{d}{d x}\left(\sin ^{4} x\right) \end{array}$$ d. Based upon your answers to parts (a)-(c), make a conjecture about \(\frac{d}{d x}\left(\sin ^{n} x\right),\) where \(n\) is a positive integer. Then prove the result by induction.

Identifying functions from an equation The following equations implicitly define one or more functions. a. Find \(\frac{d y}{d x}\) using implicit differentiation. b. Solve the given equation for \(y\) to identify the implicitly defined functions \(y=f_{1}(x), y=f_{2}(x), \ldots\) c. Use the functions found in part (b) to graph the given equation. $$y^{2}=\frac{x^{2}(4-x)}{4+x} \text { (right strophoid) }$$

Use the following table to find the given derivatives. $$\begin{array}{llllll} x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 5 & 4 & 3 & 2 & 1 \\ f^{\prime}(x) & 3 & 5 & 2 & 1 & 4 \\ g(x) & 4 & 2 & 5 & 3 & 1 \\ g^{\prime}(x) & 2 & 4 & 3 & 1 & 5 \end{array}$$ $$\left.\frac{d}{d x}\left[\frac{f(x)}{(x+2)}\right]\right|_{x=4}$$

A challenging second derivative Find \(\frac{d^{2} y}{d x^{2}},\) where \(\sqrt{y}+x y=1\).

Cobb-Douglas production function The output of an economic system \(Q,\) subject to two inputs, such as labor \(L\) and capital \(K\) is often modeled by the Cobb- Douglas production function \(Q=c L^{a} K^{b} .\) When \(a+b=1,\) the case is called constant returns to scale. Suppose \(Q=1280, a=\frac{1}{3}, b=\frac{2}{3},\) and \(c=40\) a. Find the rate of change of capital with respect to labor, \(d K / d L\) b. Evaluate the derivative in part (a) with \(L=8\) and \(K=64\)
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