Chapter 2: Q. 51 (page 210)

Calculate each of the derivatives or derivative values in Exercises 47–52.
${\frac{d^{3}}{d x^{3}} (x {(3 x^{2} + 1)}^{9})}_{x = 0}$

Short Answer

Expert verified

${\frac{d^{3}}{d x^{3}} (x {(3 x^{2} + 1)}^{9})}_{x = 0} = 162$

Step by step solution

Step 1. Given Information

We need to find ${\frac{d^{3}}{d x^{3}} (x {(3 x^{2} + 1)}^{9})}_{x = 0}$

Let us write $y = x {(3 x^{2} + 1)}^{9}$

Step 2. Finding First derivative

$\frac{d y}{d x} = {(3 x^{2} + 1)}^{9} \frac{d}{d x} x + x \frac{d}{d x} {(3 x^{2} + 1)}^{9} \frac{d y}{d x} = {(3 x^{2} + 1)}^{9} + 9 x {(3 x^{2} + 1)}^{8} (6 x) \frac{d y}{d x} = {(3 x^{2} + 1)}^{9} + 54 x^{2} {(3 x^{2} + 1)}^{8} \frac{d y}{d x} = {(3 x^{2} + 1)}^{8} (57 x^{2} + 1)$

Step 2. Finding Second Order Derivative

$\frac{d^{2} y}{d x^{2}} = 8 {(3 x^{2} + 1)}^{7} (6 x) + {(3 x^{2} + 1)}^{8} (114 x) \frac{d^{2} y}{d x^{2}} = 48 x ({(3 x^{2} + 1)}^{7}) + 114 x {(3 x^{2} + 1)}^{8} \frac{d^{2} y}{d x^{2}} = 162 x {(3 x^{2} + 1)}^{7} (19 x^{2} + 1)$

Step 4. Finding third order derivative

$\frac{d^{3} y}{d x^{3}} = 162 {(3 x^{2} + 1)}^{7} (19 x^{2} + 1) + 162 x \times 7 {(3 x^{2} + 1)}^{6} (19 x^{2} + 1) + 162 x {(3 x^{2} + 1)}^{7} (38 x) \frac{d^{3} y}{d x^{3}} = 162 {(3 x^{2} + 1)}^{6} (969 x^{4} + 102 x^{2} + 1)$

Step 5. Finding third derivative at x=0

$\frac{d^{3} y}{d x^{3}} = 162 {(0^{2} + 1)}^{6} (969 \times 0^{4} + 102 \times 0^{2} + 1) = 162 \times 1 = 162$

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Calculate each of the derivatives or derivative values in Exercises 47–52.
${\frac{d^{3}}{d x^{3}} (x {(3 x^{2} + 1)}^{9})}_{x = 0}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding First derivative

Step 2. Finding Second Order Derivative

Step 4. Finding third order derivative

Step 5. Finding third derivative at x=0

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Calculate each of the derivatives or derivative values in Exercises 47–52.d3dx3x3x2+19x=0

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding First derivative

Step 2. Finding Second Order Derivative

Step 4. Finding third order derivative

Step 5. Finding third derivative at x=0

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

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Study anywhere. Anytime. Across all devices.

Calculate each of the derivatives or derivative values in Exercises 47–52.
${\frac{d^{3}}{d x^{3}} (x {(3 x^{2} + 1)}^{9})}_{x = 0}$