Chapter 2: Q. 1 (page 208)

Differentiation review: Without using the chain rule find the derivative of each of the function f that follows some algebra may be required before differentiating
$a) f (x) = {(3 x + 1)}^{4} b) f (x) = {(\frac{1}{x} + \frac{1}{x^{2}})}^{2} c) f (x) = {(x + 1)}^{2} \sqrt{x} d) f (x) = \frac{{(x + 1)}^{2}}{\sqrt{x}}$

Short Answer

Expert verified

The derivatives can be given as

$a) f' (x) = 324 x^{3} + 324 x^{2} + 108 x + 12 b) f' (x) = \frac{- 2}{x^{3}} - \frac{6}{x^{4}} - \frac{4}{x^{5}} c) f' (x) = \frac{5}{2} x^{\frac{3}{2}} + 3 \sqrt{x} + \frac{1}{2 \sqrt{x}} d) f' (x) = \frac{3}{2} \sqrt{x} + \frac{1}{\sqrt{x}} - \frac{2}{x \sqrt{x}}$

Step by step solution

Given information

We are given several functions

Part a )Step 1: Find the derivative of the function

We get,

$f (x) = {(3 x + 1)}^{4} f (x) = 81^{4} + 108 x^{3} + 54 x^{2} + 12 x + 1 O n d i f f e r e n t i a t i n g f' (x) = 324 x^{3} + 324 x^{2} + 108 x + 12$

Which is the required answer

Part b) Step 1: Find the derivative of part b

We have,

$f (x) = {(\frac{1}{x} + \frac{1}{x^{2}})}^{2} f (x) = \frac{1}{x^{2}} + \frac{2}{x^{3}} + \frac{1}{x^{4}} O n d i f f e r e n t i a t i n g f' (x) = - \frac{2}{x^{3}} - \frac{6}{x^{4}} - \frac{4}{x^{5}}$

Part c) Step 1: Find the derivative of part c

We have,

$f (x) = {(x + 1)}^{2} \sqrt{x} f (x) = (x^{2} + 2 x + 1) \sqrt{x} f (x) = x^{\frac{5}{2}} + 2 x^{\frac{3}{2}} + \sqrt{x} O n d i f f e r e n t i a t i n g f' (x) = \frac{5}{2} x^{\frac{3}{2}} + 3 \sqrt{x} + \frac{1}{2 \sqrt{x}}$

Which is required derivative

Part d) Step 1: Find the derivative of part d

We have,

$f (x) = \frac{{(x + 1)}^{2}}{\sqrt{x}} f (x) = \frac{x^{2} + 2 x + 1}{\sqrt{x}} f (x) = x^{\frac{3}{2}} + 2 \sqrt{x} + \frac{1}{\sqrt{x}} O n d i f f e r e n t i a t i n g f' (x) = \frac{3}{2} \sqrt{x} + \frac{1}{\sqrt{x}} - \frac{2}{x \sqrt{x}}$

Which is the required derivative

Conclusion

The derivatives can be given as

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Short Answer

Step by step solution

Given information

Part a )Step 1: Find the derivative of the function

Part b) Step 1: Find the derivative of part b

Part c) Step 1: Find the derivative of part c

Part d) Step 1: Find the derivative of part d

Conclusion

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Differentiation review: Without using the chain rule find the derivative of each of the function f that follows some algebra may be required before differentiatinga)f(x)=(3x+1)4b)f(x)=(1x+1x2)2c)f(x)=(x+1)2xd)f(x)=(x+1)2x

Short Answer

Step by step solution

Given information

Part a )Step 1: Find the derivative of the function

Part b) Step 1: Find the derivative of part b

Part c) Step 1: Find the derivative of part c

Part d) Step 1: Find the derivative of part d

Conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

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Logic and Functions

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Study anywhere. Anytime. Across all devices.