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Basic definition-of-derivative calculations: Find the derivatives of the functions that follow, using (a) the h0definition of the derivative and (b) the zxdefinition of the derivative.

fx=x3

Short Answer

Expert verified

f'x=3x2

Step by step solution

01

Given Information

The given expression isfx=x3

02

Simplification

Using (a)

f'x=limh0fx+h-fxh

f'x=limh0x+h3-x3h

role="math" localid="1650970063541" f'x=limh0x3+h3+3x2h+3xh2-x3h

role="math" localid="1650970073601" f'x=limh0hh2+3x2+3xhh

role="math" localid="1650970082470" f'x=limh0h2+3x2+2xh

Applying limit

f(x)=2x2

Using (b)

role="math" localid="1650970131635" f'x=limzxfz-fxz-x

role="math" localid="1650969972120" f'x=limzxz3-x3z-x

role="math" localid="1650969988908" f'x=limzxz-xz2+x2+2zxz-x

f'x=limzxz2+x2+2zx

Applying limit

f'x=3x2

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Most popular questions from this chapter

Suppose h(t) represents the average height, in feet, of a person who is t years old.

(a) In real-world terms, what does h(12) represent and what are its units? What does h' (12) represent, and what are its units?

(b) Is h(12) positive or negative, and why? Is h'(12) positive or negative, and why?

(c) At approximately what value of t would h(t) have a maximum, and why? At approximately what value of t would h' (t) have a maximum, and why?

Use theh0definition of the derivative to prove the power rule holds for positive integers powers

Find a function that has the given derivative and value. In each case you can find the answer with an educated guess and check process it may be helpful to do some preliminary algebra

f'(x)=x(4-2x);f(0)=0

Stuart left his house at noon and walked north on Pine Street for 20minutes. At that point he realized he was late for an appointment at the dentist, whose office was located south of Stuart’s house on Pine Street; fearing he would be late, Stuart sprinted south on Pine Street, past his house, and on to the dentist’s office. When he got there, he found the office closed for lunch; he was 10minutes early for his 12:40appointment. Stuart waited at the office for 10minutes and then found out that his appointment was actually for the next day, so he walked back to his house. Sketch a graph that describes Stuart’s position over time. Then sketch a graph that describes Stuart’s velocity over time.

Find the derivatives of the functions in Exercises 21–46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

f(x)=x2-x3+5x9x-1

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