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use the definition of derivative to directly prove the differentiation rules for constant and identity function

Short Answer

Expert verified

We use definition of the derivative to prove the differentiation rules for constant and identity function

Step by step solution

01

Given information

We are given constant and identity function

02

Derivative for constant function

Consider a constant function f:RRsuch that f(x)=cfor all xR

Now we use definition of derivative

limh0f(x+h)-f(x)hlimh0c-chasforanyxwehavef(x)=c=0

03

Derivative for identity function

Consider the identity function f:RRgiven by f(x)=x

Now we apply the definition of the derivative

limh0f(x+h)-f(x)hlimh0x+h-xhlimh0hh=1

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Most popular questions from this chapter

Use the definition of the derivative to find ffor each function fin Exercises 39-54

f(x)=x3x+1

Last night Phil went jogging along Main Street. His distance from the post office t minutes after 6:00p.m. is shown in the preceding graph at the right.

(a) Give a narrative (that matches the graph) of what Phil did on his jog.

(b) Sketch a graph that represents Phil’s instantaneous velocity t minutes after 6:00p.m. Make sure you label the tick marks on the vertical axis as accurately as you can.

(c) When was Phil jogging the fastest? The slowest? When was he the farthest away from the post office? The closest to the post office?

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)=2-3x+1

Use the definition of the derivative to find the equations of the lines described in Exercises 59-64.

The line that passes through the point (3,2)and is parallel to the tangent line to f(x)=1x at x=-1.

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.

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