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Chapter 2: Q 26 (page 238)

Find the derivatives of the function
$f (x) = \frac{\sin h^{- 1} x}{\tan h^{- 1} x}$

Short Answer

Expert verified

$f' (x) = \frac{d y}{d x} = \frac{1}{\sqrt{x^{2} - 1}}$

Step by step solution

01

Step 1:Given information

The given expression is $f (x) = \frac{\sin h^{- 1} x}{\tan h^{- 1} x}$

02

Step 2:Simplification

Given

$f (x) = y = \frac{\sin h^{- 1} x}{\tan h^{- 1} x}$

$鈬� y = \frac{\sin h^{- 1} x}{\frac{\sin h^{- 1} x}{\cos h^{- 1} x}}$

$鈬� y = \sin h^{- 1} x . \frac{\cos h^{- 1} x}{\sin h^{- 1} x}$

$鈬� y = \cos h^{- 1} x$

Differentiating w.r.t x

$鈬� \frac{d y}{d x} = \frac{d}{d x} (\cos h^{- 1} x)$

$鈬� \frac{d y}{d x} = \frac{1}{\sqrt{x^{2} - 1}}$

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Most popular questions from this chapter

Write down a rule for differentiating a composition $f (u (v (w (x))))$ of four functions

(a) in 鈥減rime鈥� notation and

(b) in Leibniz notation.

For each function $f$ graphed in Exercises 65-68, determine the values of $x$ at which $f$ fails to be continuous and/or differentiable. At such points, determine any left or right continuity or differentiability. Sketch secant lines supporting your answers.

Velocity $v (t)$ is the derivative of position $s (t)$ . It is also true that acceleration $a (t)$ (the rate of change of velocity) is the derivative of velocity. If a race car鈥檚 position in miles t hours after the start of a race is given by the function $s (t)$ , what are the units of $s (1.2)$ ? What are the units and real-world interpretation of $v (1.2)$ ? What are the units and real-world interpretations of $a (1.2)$ ?

Find the derivatives of the functions in Exercises 21鈥�46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

$f (x) = x^{- \frac{1}{2}} {(x^{2} - 1)}^{3}$

Find the derivatives of the functions in Exercises 21鈥�46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

$f (x) = \frac{1}{\sqrt{x^{2} + 1}}$

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