**Differentiate the Following Polynomials***Note: some questions may not include all the possible terms.*Given polynomial:\[-x^5 - 5x^4 + 5x^3 - 11x^2 - 21x - 40\]Calculate the first derivative:\[ f'(x) = \text{[First Derivative]} \]Calculate the second derivative:\[ f''(x) = \text{[Second Derivative]} \] Fill in the boxes with the derivatives as required.

Answered: -25 – 5x4 + 5x³ – 11æ² – 21x – 40

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Step 1

Given:

$f (x) = - x^{5} - 5 x^{4} + 5 x^{3} - 11 x^{2} - 21 x - 40$

To find:

$f' (x)$ and $f'' (x)$ .

Step 2

Differentiate f(x) with respect to x it yields f'(x),

Thus,

$\frac{d}{d x} f (x) = \frac{d}{d x} (- x^{5} - 5 x^{4} + 5 x^{3} - 11 x^{2} - 21 x - 40)$

$f' (x) = \frac{d}{d x} (- x^{5} - 5 x^{4} + 5 x^{3} - 11 x^{2} - 21 x - 40)$

$f' (x) = \frac{d}{d x} (- x^{5}) + \frac{d}{d x} (- 5 x^{4}) + \frac{d}{d x} (5 x^{3}) - \frac{d}{d x} (11 x^{2}) - \frac{d}{d x} (21 x) - \frac{d}{d x} (40)$

Using the power rule $\frac{d}{d x} x^{n} = n x^{n - 1}$ ,

Thus, above equation become,

$f' (x) = (- 5 x^{5 - 1} - 5 (4) x^{4 - 1} + 5 (3) x^{3 - 1} - 11 (2) x^{2 - 1} - 21)$

On simplifying,

$f' (x) = (- 5 x^{4} - 5 (4) x^{3} + 5 (3) x^{2} - 11 (2) x^{1} - 21)$

f' (x) = (- 5 x^{4} - 20 x^{3} + 15 x^{2} - 22 x^{1} - 21)

Which is required.

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