Q. 1 Differentiation review: Without ... [FREE SOLUTION]

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

Decision Maths

Probability and Statistics

Applied Mathematics

Discrete Mathematics

Calculus

Geometry

Study anywhere. Anytime. Across all devices.