Q. 48 Find the derivatives of each of ... [FREE SOLUTION]

Chapter 2: Q. 48 (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
$f (x) = x^{- 2} e^{4 x} \sin^{- 1} x$

Short Answer

Expert verified

The derivative is $\frac{4 e^{4 x} \sin^{- 1} x}{x^{2}} + \frac{e^{4 x}}{x^{2} \sqrt{1 - x^{2}}} - \frac{2 e^{4 x} \sin^{- 1} x}{x^{3}}$ .

Step by step solution

Step 1. Given Information.

The given function is $f (x) = x^{- 2} e^{4 x} \sin^{- 1} x$ .

Step 2. Preliminary Algebra.

We know,

Quotient Rule:

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

Product rule:

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

Chain Rule:

$(f 鈭� u)^{'} (x) = f^{'} (u (x)) u^{'} (x)$

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

$\frac{d}{d x} e^{x} = e^{x}$

Step 3. Derivative of the function.

The derivative of the function will be,

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

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91影视

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
$f (x) = x^{- 2} e^{4 x} \sin^{- 1} x$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Preliminary Algebra.

Step 3. Derivative of the function.

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91影视

Find the derivatives of each of the functions in Exercises 17鈥�50. In some cases it may be convenient to do some preliminary algebraf(x)=x-2e4xsin-1x

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

Mechanics Maths

Discrete Mathematics

Pure Maths

Theoretical and Mathematical Physics

Probability and Statistics

Decision Maths

Study anywhere. Anytime. Across all devices.

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
$f (x) = x^{- 2} e^{4 x} \sin^{- 1} x$