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Chapter 7: Problem 36

Find the derivatives of the following functions. $$f(u)=\sinh ^{-1}(\tan u)$$

Short Answer

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Question: Find the derivative of the function $$f(u)=\sinh^{-1}(\tan u)$$. Answer: The derivative of $$f(u)=\sinh^{-1}(\tan u)$$ is $$\frac{d}{du}(f(u)) = \frac{1+\tan^2 u}{\sqrt{1+(\tan u)^2}}$$.

Step by step solution

01

Finding the derivative of g(u) = tan u

To find the derivative of $$g(u)=\tan u$$, recall that: $$\frac{d}{du}(\tan u) = \sec^2 u$$ So, the derivative of $$g(u)$$ is $$g'(u)=\sec^2 u$$.
02

Finding the derivative of h(v) = sinh^(-1)(v)

To find the derivative of $$h(v)=\sinh^{-1}(v)$$, recall that: $$\frac{d}{dv}(\sinh^{-1}(v)) = \frac{1}{\sqrt{1+v^2}}$$ So, the derivative of $$h(v)$$ is $$h'(v)=\frac{1}{\sqrt{1+v^2}}$$.
03

Applying the chain rule

Now that we have the derivatives of both the inner and outer functions, we can apply the chain rule. Recall that: $$(f \circ g)'(x) = f'(g(x)) \cdot g'(x)$$ So, plugging our functions into the chain rule, we get: $$\frac{d}{du}(f(u)) = h'(\tan u) \cdot \sec^2 u$$ To compute the final expression, substitute the expressions for $$h'(v)$$ and $$g'(u)$$ from steps 1 and 2: $$\frac{d}{du}(f(u)) = \frac{1}{\sqrt{1+(\tan u)^2}} \cdot \sec^2 u$$
04

Simplifying the expression

Recall that: $$\sec^2 u = 1 + \tan^2 u$$ So, we can simplify our expression as follows: $$\frac{d}{du}(f(u)) = \frac{1}{\sqrt{1+(\tan u)^2}} \cdot (1+\tan^2 u) = \frac{1+\tan^2 u}{\sqrt{1+(\tan u)^2}}$$ Thus, the derivative of $$f(u)=\sinh^{-1}(\tan u)$$ is: $$\frac{d}{du}(f(u)) = \frac{1+\tan^2 u}{\sqrt{1+(\tan u)^2}}$$

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Key Concepts

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

Chain Rule in Calculus
The chain rule is a fundamental tool in calculus that allows us to find the derivative of a composite function. A composite function is essentially a function within another function, which can be notated as \(f \(g(x)\)\).

To apply the chain rule, we need to differentiate the outer function (in this case, \(f\)) with respect to the inner function \(g\), and then multiply by the derivative of the inner function \(g\) with respect to the variable \(x\). The chain rule can be formally written as:
\[\frac{d}{dx}\big(f\big(g(x)\big)\big) = f'\big(g(x)\big) \cdot g'(x)\]
This tool is incredibly useful when dealing with functions such as the inverse hyperbolic sine of the tangent function, where one function is nested inside another. By breaking down the differentiation process into manageable steps and using the chain rule, we make the process of finding complicated derivatives much more straightforward.
Derivative of Tangent Function
The tangent function, denoted as \(\tan(x)\), is a trigonometric function that can be understood as the ratio of the sine and cosine functions. The derivative of this function is important because it not only helps in solving differential equations but also in various applications of physics and engineering.

When finding the derivative of the tangent function, we apply basic differentiation rules that give us:
\[\frac{d}{dx}(\tan x) = \sec^2 x\]
This result emerges from the quotient rule applied to \(\sin(x)\) and \(\cos(x)\). The derivative \(\sec^2 x\) is significant because it appears often in problems involving trigonometric functions. Being familiar with it allows students to solve these problems efficiently.
Derivative of Inverse Hyperbolic Sine
The inverse hyperbolic sine function, often written as \(\sinh^{-1}(x)\) or \(\text{asinh}(x)\), is one of the inverse hyperbolic functions which mirrors the properties of inverse trigonometric functions. Like its trigonometric counterparts, finding derivatives of inverse hyperbolic functions includes applying the definitions and rules of differentiation to these special functions.

The derivative of the inverse hyperbolic sine of \(v\) is given by the formula:
\[\frac{d}{dv}(\sinh^{-1}(v)) = \frac{1}{\sqrt{1+v^2}}\]
This formula is derived from the derivative of the hyperbolic sine function, considering the inverse function theorem. Understanding the derivatives of inverse hyperbolic functions is crucial for students, especially when they encounter these functions composed with others, in which case the chain rule is applied for differentiation.

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Most popular questions from this chapter

Power lines A power line is attached at the same height to two utility poles that are separated by a distance of \(100 \mathrm{ft}\); the power line follows the curve \(f(x)=a \cosh \frac{x}{a} .\) Use the following steps to find the value of \(a\) that produces a sag of \(10 \mathrm{ft}\) midway between the poles. Use a coordinate system that places the poles at \(x=\pm 50\). a. Show that \(a\) satisfies the equation \(\cosh \frac{50}{a}-1=\frac{10}{a}\) b. Let \(t=\frac{10}{a},\) confirm that the equation in part (a) reduces to \(\cosh 5 t-1=t,\) and solve for \(t\) using a graphing utility. Report your answer accurate to two decimal places. c. Use your answer in part (b) to find \(a\), and then compute the length of the power line.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. \(\frac{d}{d x}(\sinh (\ln 3))=\frac{\cosh (\ln 3)}{3}\) b. \(\frac{d}{d x}(\sinh x)=\cosh x\) and \(\frac{d}{d x}(\cosh x)=-\sinh x\) c. \(\ln (1+\sqrt{2})=-\ln (-1+\sqrt{2})\) d. \(\int_{0}^{1} \frac{d x}{4-x^{2}}=\frac{1}{2}\left(\operatorname{coth}^{-1} \frac{1}{2}-\operatorname{coth}^{-1} 0\right)\)

Points of inflection Find the \(x\) -coordinate of the point(s) of inflection of \(f(x)=\tanh ^{2} x\)

Designing exponential decay functions Devise an exponential decay function that fits the following data; then answer the accompanying questions. Be sure to identify the reference point \((t=0)\) and units of time. Crime rate The homicide rate decreases at a rate of \(3 \% / \mathrm{yr}\) in a city that had 800 homicides/yr in 2018 . At this rate, when will the homicide rate reach 600 homicides/yr?

Compounded inflation The U.S. government reports the rate of inflation (as measured by the consumer price index) both monthly and annually. Suppose for a particular month, the monthly rate of inflation is reported as \(0.8 \% .\) Assuming this rate remains constant, what is the corresponding annual rate of inflation? Is the annual rate 12 times the monthly rate? Explain.
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