Chapter 2: Q. 37 (page 233)

Find the derivatives of each of the functions in Exercises 17–50. In some cases it may be convenient to do some preliminary algebra
$f (x) = \sin^{- 1} (3 x^{2})$

Short Answer

Expert verified

The derivative of the function is $\frac{6 x}{\sqrt{1 - 9 x^{4}}} .$

Step by step solution

Step 1. Given Information.

The given function is $f (x) = \sin^{- 1} (3 x^{2})$ .

Step 2. Preliminary algebra .

The Derivative of products of functions is,

$(f g)^{'} (x) = f^{'} (x) g (x) + f (x) g^{'} (x)$

The Derivative of Quotients of functions is,

${(\frac{f}{g})}^{'} (x) = \frac{f^{'} (x) g (x) - f (x) g^{'} (x)}{(g (x))^{2}}$

Chain rule states that,

$(f \circ u)^{'} (x) = f^{'} (u (x)) u^{'} (x)$

Also,

$\frac{d}{d x} (\sin x) = \cos x \frac{d}{d x} (\cos x) = - \sin x \frac{d}{d x} (\tan x) = \sec^{2} x \frac{d}{d x} (\sec x) = \sec x \tan x \frac{d}{d x} (\cot x) = - \csc^{2} x \frac{d}{d x} (\csc x) = - \csc x \cot x \frac{d}{d x} (\sin^{- 1} x) = \frac{1}{\sqrt{1 - x^{2}}} \frac{d}{d x} (\tan^{- 1} x) = \frac{1}{1 + x^{2}} \frac{d}{d x} (\sec^{- 1} x) = \frac{1}{| x | \sqrt{x^{2} - 1}}$

Step 3. Derivative of the function.

The derivative of the function will be,

$\frac{d}{d x} (\sin^{- 1} (3 x^{2})) = \frac{1}{\sqrt{1 - {(3 x^{2})}^{2}}} (\frac{d}{d x} (3 x^{2})) = \frac{1}{\sqrt{1 - 9 x^{4}}} (3 (2 x)) = \frac{1}{\sqrt{1 - 9 x^{4}}} (6 x) = \frac{6 x}{\sqrt{1 - 9 x^{4}}}$

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Find the derivatives of each of the functions in Exercises 17–50. In some cases it may be convenient to do some preliminary algebra
$f (x) = \sin^{- 1} (3 x^{2})$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Preliminary algebra .

Step 3. Derivative of the function.

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Find the derivatives of each of the functions in Exercises 17–50. In some cases it may be convenient to do some preliminary algebraf(x)=sin-13x2

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Preliminary algebra .

Step 3. Derivative of the function.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

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Study anywhere. Anytime. Across all devices.

Find the derivatives of each of the functions in Exercises 17–50. In some cases it may be convenient to do some preliminary algebra
$f (x) = \sin^{- 1} (3 x^{2})$