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

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:

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

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

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

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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Find the derivatives of each of the functions in Exercises 17–50. In some cases it may be convenient to do some preliminary algebra
$f (x) = x^{- 2} e^{4 x} \sin^{- 1} x$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Preliminary Algebra.

Step 3. Derivative of the function.

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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)=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

Statistics

Probability and Statistics

Logic and Functions

Calculus

Theoretical and Mathematical Physics

Applied Mathematics

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