Q. 40 Find the derivatives of the func... [FREE SOLUTION]

Chapter 2: Q. 40 (page 210)

Find the derivatives of the functions in Exercises 21鈥�46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.
$f (x) = \frac{(3 x + 1) {(x^{4} - 3)}^{4}}{{(x + 5)}^{- 2} {(1 + x^{2})}^{5}}$

Short Answer

Expert verified

The required answer is $\frac{(3 {(x^{4} - 3)}^{4} + 16 x^{3} (3 x + 1) {(x^{4} - 3)}^{3}) {(x + 5)}^{- 2} {(1 + x^{2})}^{5} - (3 x + 1) {(x^{4} - 3)}^{4} (- 2 {(x + 5)}^{- 3} {(1 + x^{2})}^{5} + 10 x {(x + 5)}^{- 2} {(1 + x^{2})}^{4})}{{(x + 5)}^{- 1} {(1 + x^{2})}^{\frac{5}{2}}}$

Step by step solution

Step 1. Given Information

The given function is $f (x) = \frac{(3 x + 1) {(x^{4} - 3)}^{4}}{{(x + 5)}^{- 2} {(1 + x^{2})}^{5}}$

Step 2. Calculation

Differentiate both the sides with respect to x, we get,

$f' (x) = \frac{(\frac{d}{d x} (3 x + 1) {(x^{4} - 3)}^{4}) {(x + 5)}^{- 2} {(1 + x^{2})}^{5} - (3 x + 1) {(x^{4} - 3)}^{4} (\frac{d}{d x} {(x + 5)}^{- 2} {(1 + x^{2})}^{5})}{{({(x + 5)}^{- 2} {(1 + x^{2})}^{5})}^{\frac{1}{2}}} = \frac{(3 {(x^{4} - 3)}^{4} + (3 x + 1)) (4 {(x^{4} - 3)}^{3} \frac{d}{d x} (x^{4} - 3)) {(x + 5)}^{- 2} {(1 + x^{2})}^{5} - (3 x + 1) {(x^{4} - 3)}^{4} (- 2 {(x + 5)}^{- 2 - 1} \frac{d}{d x} (x + 5)) {(1 + x^{2})}^{5} + {(x + 5)}^{- 2} (5 {(1 + x^{2})}^{4} \frac{d}{d x} (1 + x^{2}))}{{({(x + 5)}^{- 2} {(1 + x^{2})}^{5})}^{\frac{1}{2}}} = \frac{(3 {(x^{4} - 3)}^{4} + (3 x + 1)) (4 {(x^{4} - 3)}^{3} (4 x^{3})) {(x + 5)}^{- 2} {(1 + x^{2})}^{5} - (3 x + 1) {(x^{4} - 3)}^{4} (- 2 {(x + 5)}^{- 3}) {(1 + x^{2})}^{5} + {(x + 5)}^{- 2} (5 {(1 + x^{2})}^{4} (2 x))}{{({(x + 5)}^{- 2} {(1 + x^{2})}^{5})}^{\frac{1}{2}}} = \frac{(3 {(x^{4} - 3)}^{4} + 16 x^{3} (3 x + 1) {(x^{4} - 3)}^{3}) {(x + 5)}^{- 2} {(1 + x^{2})}^{5} - (3 x + 1) {(x^{4} - 3)}^{4} (- 2 {(x + 5)}^{- 3} {(1 + x^{2})}^{5} + 10 x {(x + 5)}^{- 2} {(1 + x^{2})}^{4})}{{(x + 5)}^{- 1} {(1 + x^{2})}^{\frac{5}{2}}}$

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Find the derivatives of the functions in Exercises 21鈥�46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.
$f (x) = \frac{(3 x + 1) {(x^{4} - 3)}^{4}}{{(x + 5)}^{- 2} {(1 + x^{2})}^{5}}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculation

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91影视

Find the derivatives of the functions in Exercises 21鈥�46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.f(x)=3x+1x4-34x+5-21+x25

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculation

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Calculus

Mechanics Maths

Geometry

Theoretical and Mathematical Physics

Statistics

Study anywhere. Anytime. Across all devices.

Find the derivatives of the functions in Exercises 21鈥�46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.
$f (x) = \frac{(3 x + 1) {(x^{4} - 3)}^{4}}{{(x + 5)}^{- 2} {(1 + x^{2})}^{5}}$