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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 differentiating

a)f(x)=(3x+1)4b)f(x)=(1x+1x2)2c)f(x)=(x+1)2xd)f(x)=(x+1)2x

Short Answer

Expert verified

The derivatives can be given as

a)f'(x)=324x3+324x2+108x+12b)f'(x)=-2x3-6x4-4x5c)f'(x)=52x32+3x+12xd)f'(x)=32x+1x-2xx

Step by step solution

01

Given information

We are given several functions

02

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

We get,

f(x)=(3x+1)4f(x)=814+108x3+54x2+12x+1Ondifferentiatingf'(x)=324x3+324x2+108x+12

Which is the required answer

03

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

We have,

f(x)=(1x+1x2)2f(x)=1x2+2x3+1x4Ondifferentiatingf'(x)=-2x3-6x4-4x5

04

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

We have,

f(x)=(x+1)2x f(x)=(x2+2x+1)xf(x)=x52+2x32+xOndifferentiatingf'(x)=52x32+3x+12x

Which is required derivative

05

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

We have,

f(x)=(x+1)2xf(x)=x2+2x+1xf(x)=x32+2x+1xOndifferentiatingf'(x)=32x+1x-2xx

Which is the required derivative

06

Conclusion

The derivatives can be given as

a)f'(x)=324x3+324x2+108x+12b)f'(x)=-2x3-6x4-4x5c)f'(x)=52x32+3x+12xd)f'(x)=32x+1x-2xx

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Most popular questions from this chapter

Differentiate f(x)=3x+x2in three ways. When you have completed all three parts, show that your three answers are the same:

(a) with the chain rule

(b) with the product rule but not the chain rule

(c) without the chain or product rules.

Velocity v(t) is the derivative of position s(t). It is also true that acceleration a(t) (the rate of change of velocity) is the derivative of velocity. If a race carโ€™s position in miles t hours after the start of a race is given by the function s(t), what are the units of s(1.2)? What are the units and real-world interpretation of v(1.2)? What are the units and real-world interpretations of a(1.2)?

For each function graphed in Exercises 65-68, determine the values of xat which ffails to be continuous and/or differentiable. At such points, determine any left or right continuity or differentiability. Sketch secant lines supporting your answers.

For each function fgraphed in Exercises 65-68, determine the values of xat which ffails to be continuous and/or differentiable. At such points, determine any left or right continuity or differentiability. Sketch secant lines supporting your answers.

A tomato plant given xounces of fertilizer will successfully bear T(x)pounds of tomatoes in a growing season.

(a) In real-world terms, what does T(5)represent and what are its units? What does T'(5)represent and what are its units?

(b) A study has shown that this fertilizer encourages tomato production when less than 20ounces are used, but inhibits production when more than 20ounces are used. When is T(x)positive and when is T(x)negative? When is T'(x)positive and when is T'(x)negative?

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