Q. 50 Calculate each of the derivative... [FREE SOLUTION]

Chapter 2: Q. 50 (page 210)

Calculate each of the derivatives or derivative values in Exercises 47鈥�52.
$\frac{d^{2}}{d x^{2}} ({(5 x^{4} - 3 x^{2})}^{7} (2 x^{3} + 1))$

Short Answer

Expert verified

$\frac{d^{2}}{d x^{2}} ({(5 x^{4} - 3 x^{2})}^{7} (2 x^{3} + 1)) = 6 x^{12} {(5 x^{2} - 3)}^{5} (7750 x^{7} - 5170 x^{5} + 3150 x^{4} + 816 x^{3} - 1925 x^{2} + 273)$

Step by step solution

Step 1. Given Information

We asks to find the second order derivative of $\frac{d^{2}}{d x^{2}} ({(5 x^{4} - 3 x^{2})}^{7} (2 x^{3} + 1))$

Let us write $y = ({(5 x^{4} - 3 x^{2})}^{7} (2 x^{3} + 1))$

Step 1. Finding First order derivative

$\frac{d y}{d x} = (2 x^{3} + 1) \frac{d}{d x} {(5 x^{4} - 3 x^{2})}^{7} + {(5 x^{4} - 3 x^{2})}^{7} \frac{d}{d x} (2 x^{3} + 1) \frac{d y}{d x} = 7 (2 x^{3} + 1) {(5 x^{4} - 3 x^{2})}^{6} \frac{d}{d x} (5 x^{4} - 3 x^{2}) + {(5 x^{4} - 3 x^{2})}^{7} (6 x^{2}) \frac{d y}{d x} = 2 x^{13} {(5 x^{2} - 3)}^{6} (155 x^{5} - 51 x^{3} + 70 x^{2} - 21)$

Step 3. Finding Second Order Derivative

$\frac{d^{2} y}{d x^{2}} = {(5 x^{2} - 3)}^{6} (155 x^{5} - 51 x^{3} + 70 x^{2} - 21) \frac{d}{d x} 2 x^{13} + 2 x^{13} (155 x^{5} - 51 x^{3} + 70 x^{2} - 21) \frac{d}{d x} {(5 x^{2} - 3)}^{6} + 2 x^{13} {(5 x^{2} - 3)}^{6} \frac{d}{d x} (155 x^{5} - 51 x^{3} + 70 x^{2} - 21) \frac{d^{2} y}{d x^{2}} = 26 x^{12} {(5 x^{2} - 3)}^{6} (155 x^{5} - 51 x^{3} + 70 x^{2} - 21) + 12 x^{13} (10 x - 0) (155 x^{5} - 51 x^{3} + 70 x^{2} - 21) + 2 x^{13} {(5 x^{2} - 3)}^{6} (775 x^{5} - 153 x^{2} + 140 x - 0) \frac{d^{2} y}{d x^{2}} = 6 x^{12} {(5 x^{2} - 3)}^{5} (7750 x^{7} - 5170 x^{5} + 3150 x^{4} + 816 x^{3} - 1925 x^{2} + 273)$

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91影视

Calculate each of the derivatives or derivative values in Exercises 47鈥�52.
$\frac{d^{2}}{d x^{2}} ({(5 x^{4} - 3 x^{2})}^{7} (2 x^{3} + 1))$

Short Answer

Step by step solution

Step 1. Given Information

Step 1. Finding First order derivative

Step 3. Finding Second Order Derivative

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91影视

Calculate each of the derivatives or derivative values in Exercises 47鈥�52.d2dx25x4-3x272x3+1

Short Answer

Step by step solution

Step 1. Given Information

Step 1. Finding First order derivative

Step 3. Finding Second Order Derivative

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Logic and Functions

Theoretical and Mathematical Physics

Decision Maths

Mechanics Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Calculate each of the derivatives or derivative values in Exercises 47鈥�52.
$\frac{d^{2}}{d x^{2}} ({(5 x^{4} - 3 x^{2})}^{7} (2 x^{3} + 1))$