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Chapter 2: Q. 50 (page 233)

Find the derivatives of each of the functions in Exercises 17鈥�50. In some cases it may be convenient to do some preliminary algebra
$f (x) = \frac{\sin (\arcsin x)}{\arctan x}$

Short Answer

Expert verified

The derivative is $\frac{1}{\arctan x} - \frac{x}{(1 + x^{2}) \arctan^{2} x}$ .

Step by step solution

01

Step 1. Given Information.

The given function is $f (x) = \frac{\sin (\arcsin x)}{\arctan x}$

02

Step 2. Preliminary Algebra.

We know,

$\frac{d}{d x} (\sin^{- 1} x) = \frac{1}{\sqrt{1 - x^{2}}} \frac{d}{d x} (\tan^{- 1} x) = \frac{1}{1 + x^{2}} C h a i n R u l e : (f 鈭� u)^{'} (x) = f^{'} (u (x)) u^{'} (x)$

03

Step 3. Derivative of the function.

The derivative of the function is,

$\frac{d}{d x} (\frac{\sin (\arcsin x)}{\arctan x}) = \frac{d}{d x} (\frac{x}{\arctan x}) = \frac{(\frac{d}{d x} x) \arctan x - x (\frac{d}{d x} \arctan x)}{(\arctan x)^{2}} = \frac{\arctan x - (x) \frac{1}{1 + x^{2}}}{(\arctan x)^{2}} = \frac{\arctan x - \frac{x}{1 + x^{2}}}{(\arctan x)^{2}} = \frac{1}{\arctan x} - \frac{x}{(1 + x^{2}) \arctan^{2} x}$

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Most popular questions from this chapter

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.

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