Q. 1 Finding the second derivative: F... [FREE SOLUTION]

Chapter 3: Q. 1 (page 274)

Finding the second derivative: For each of the following functions $f$ , calculate and simplify the second derivative $f''$ .
1. $f (x) = \frac{x - 1}{3 x^{2} - 4 x + 2} .$
2. $f (x) = \frac{x}{\sqrt{x^{2} + 1}} .$
3. $f (x) = \frac{1 - x}{e^{x}} .$
4. $f (x) = e^{3 x} \ln (x^{2} + 1) .$

Short Answer

Expert verified

$1 . f'' (x) = \frac{2 (9 x^{3} - 27 x^{2} + 18 x - 2)}{{(3 x^{2} - 4 x + 2)}^{3}} .$

$2 . f'' (x) = \frac{- 3 x}{{(\sqrt{x^{2} + 1})}^{5}} .$

$3 . f'' (x) = \frac{3 - x}{e^{x}} .$

$4 . f'' (x) = 3 (\frac{2 x e^{3 x}}{x^{2} + 1} + 3 e^{3 x} \ln (x^{2} + 1)) + \frac{2 e^{3 x} (3 x^{3} - x^{2} + 3 x + 1)}{{(x^{2} + 1)}^{2}} .$

Step by step solution

Step 1. Given Information.

Given the function:

1. $f (x) = \frac{x - 1}{3 x^{2} - 4 x + 2} .$

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

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

4. $f (x) = e^{3 x} \ln (x^{2} + 1) .$

Step 2. Calculating the first derivative part(a).

$f (x) = \frac{x - 1}{3 x^{2} - 4 x + 2} f' (x) = \frac{(3 x^{2} - 4 x + 2) \frac{d}{d x} (x - 1) - (x - 1) \frac{d}{d x} (3 x^{2} - 4 x + 2)}{{(3 x^{2} - 4 x + 2)}^{2}} f' (x) = \frac{(3 x^{2} - 4 x + 2) - (x - 1) (6 x - 4)}{{(3 x^{2} - 4 x + 2)}^{2}} f' (x) = \frac{(3 x^{2} - 4 x + 2) - (6 x^{2} - 10 x + 4)}{{(3 x^{2} - 4 x + 2)}^{2}} f' (x) = \frac{(- 3 x^{2} + 6 x - 2)}{{(3 x^{2} - 4 x + 2)}^{2}}$

Step 3. Calculating the second derivative part(a).

$N o w d i f f e r e n t i a t i n g o n c e a g a i n, f'' (x) = \frac{({(3 x^{2} - 4 x + 2)}^{2}) \frac{d}{d x} (- 3 x^{2} + 6 x - 2) - (- 3 x^{2} + 6 x - 2) \frac{d}{d x} ({(3 x^{2} - 4 x + 2)}^{2})}{{(3 x^{2} - 4 x + 2)}^{4}} f'' (x) = \frac{({(3 x^{2} - 4 x + 2)}^{2}) (- 6 x + 6) - (- 3 x^{2} + 6 x - 2) (2 (3 x^{2} - 4 x + 2)) (6 x - 4)}{{(3 x^{2} - 4 x + 2)}^{4}} f'' (x) = \frac{((3 x^{2} - 4 x + 2)) (- 6 x + 6) - (- 3 x^{2} + 6 x - 2) (2) (6 x - 4)}{{(3 x^{2} - 4 x + 2)}^{3}} f'' (x) = \frac{2 (9 x^{3} - 27 x^{2} + 18 x - 2)}{{(3 x^{2} - 4 x + 2)}^{3}} .$

Step 4. Calculating the first derivative part(b).

$f (x) = \frac{x}{\sqrt{x^{2} + 1}} f' (x) = \frac{(\sqrt{x^{2} + 1}) \frac{d}{d x} (x) - (x) \frac{d}{d x} (\sqrt{x^{2} + 1})}{x^{2} + 1} f' (x) = \frac{(\sqrt{x^{2} + 1}) - (x) \frac{2 x}{2 \sqrt{x^{2} + 1}}}{x^{2} + 1} f' (x) = \frac{(\sqrt{x^{2} + 1}) - \frac{x^{2}}{\sqrt{x^{2} + 1}}}{x^{2} + 1} f' (x) = \frac{(x^{2} + 1) - x^{2}}{{(\sqrt{x^{2} + 1})}^{3}} f' (x) = \frac{1}{{(\sqrt{x^{2} + 1})}^{3}} .$

Step 5. Calculating the second derivative part(b).

$f' (x) = \frac{1}{{(\sqrt{x^{2} + 1})}^{3}} = {(x^{2} + 1)}^{- \frac{3}{2}} D i f f e r e n t i a t i n g o n c e a g a i n w e g e t, f'' (x) = - \frac{3}{2} {(x^{2} + 1)}^{- \frac{5}{2}} (2 x) = \frac{- 3 x}{{(\sqrt{x^{2} + 1})}^{5}} .$

Step 6. Calculating the first derivative part(c).

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

Step 7. Calculating the second derivative part(c).

$f' (x) = \frac{x - 2}{e^{x}} D i f f e r e n t i a t i n g o n c e a g a i n, f'' (x) = \frac{e^{x} \frac{d}{d x} (x - 2) - (x - 2) \frac{d}{d x} (e^{x})}{e^{2 x}} f'' (x) = \frac{e^{x} - (x - 2) (e^{x})}{e^{2 x}} f'' (x) = \frac{1 - (x - 2)}{e^{x}} = \frac{3 - x}{e^{x}} .$

Step 8. Calculating the first derivative part(d).

$f (x) = e^{3 x} \ln (x^{2} + 1) f' (x) = e^{3 x} \frac{d}{d x} (\ln (x^{2} + 1)) + \ln (x^{2} + 1) \frac{d}{d x} (e^{3 x}) f' (x) = e^{3 x} \frac{2 x}{x^{2} + 1} + (\ln (x^{2} + 1)) (3 e^{3 x}) f' (x) = \frac{2 x e^{3 x}}{x^{2} + 1} + 3 f (x) .$

Step 9. Calculating the second derivative part(d).

$f' (x) = \frac{2 x e^{3 x}}{x^{2} + 1} + 3 f (x) D i f f e r e n t i a t i n g o n c e a g a i n, a s w e k n o w t h e d e r i v a t i v e o f f (x) w e w i l l f i n d o n l y t h e f i r s t t e r m, f'' (x) = 3 f' (x) + \frac{(x^{2} + 1) \frac{d}{d x} (2 x e^{3 x}) - 2 x e^{3 x} \frac{d}{d x} (x^{2} + 1)}{{(x^{2} + 1)}^{2}} f'' (x) = 3 f' (x) + \frac{(x^{2} + 1) (2 e^{3 x} + 2 x 3 e^{3 x}) - 2 x e^{3 x} (2 x)}{{(x^{2} + 1)}^{2}} f'' (x) = 3 f' (x) + \frac{2 e^{3 x} (x^{2} + 1) (1 + 3 x) - 4 x^{2} e^{3 x}}{{(x^{2} + 1)}^{2}} f'' (x) = 3 f' (x) + \frac{2 e^{3 x} ((3 x^{3} + x^{2} + 3 x + 1) - 2 x^{2})}{{(x^{2} + 1)}^{2}} f'' (x) = 3 (\frac{2 x e^{3 x}}{x^{2} + 1} + 3 e^{3 x} \ln (x^{2} + 1)) + \frac{2 e^{3 x} (3 x^{3} - x^{2} + 3 x + 1)}{{(x^{2} + 1)}^{2}} .$

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91影视

Finding the second derivative: For each of the following functions $f$ , calculate and simplify the second derivative $f''$ .
1. $f (x) = \frac{x - 1}{3 x^{2} - 4 x + 2} .$
2. $f (x) = \frac{x}{\sqrt{x^{2} + 1}} .$
3. $f (x) = \frac{1 - x}{e^{x}} .$
4. $f (x) = e^{3 x} \ln (x^{2} + 1) .$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Calculating the first derivative part(a).

Step 3. Calculating the second derivative part(a).

Step 4. Calculating the first derivative part(b).

Step 5. Calculating the second derivative part(b).

Step 6. Calculating the first derivative part(c).

Step 7. Calculating the second derivative part(c).

Step 8. Calculating the first derivative part(d).

Step 9. Calculating the second derivative part(d).

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91影视

Finding the second derivative: For each of the following functions f, calculate and simplify the second derivative f''.1. f(x)=x-13x2-4x+2.2. f(x)=xx2+1.3. f(x)=1-xex.4.f(x)=e3xlnx2+1.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Calculating the first derivative part(a).

Step 3. Calculating the second derivative part(a).

Step 4. Calculating the first derivative part(b).

Step 5. Calculating the second derivative part(b).

Step 6. Calculating the first derivative part(c).

Step 7. Calculating the second derivative part(c).

Step 8. Calculating the first derivative part(d).

Step 9. Calculating the second derivative part(d).

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Mechanics Maths

Calculus

Applied Mathematics

Theoretical and Mathematical Physics

Geometry

Study anywhere. Anytime. Across all devices.

Finding the second derivative: For each of the following functions $f$ , calculate and simplify the second derivative $f''$ .
1. $f (x) = \frac{x - 1}{3 x^{2} - 4 x + 2} .$
2. $f (x) = \frac{x}{\sqrt{x^{2} + 1}} .$
3. $f (x) = \frac{1 - x}{e^{x}} .$
4. $f (x) = e^{3 x} \ln (x^{2} + 1) .$