Question

Calculus:

Find the derivative by the limit process for f(x)=3x+2

Answer 1

The definition of derivative is:
\[\lim_{x \rightarrow a} \frac{ f(x)-f(a) }{ x-a }\]

Answer 2

lim 3x+2-f(a)
x-->a x-a

Answer 3

\[\lim_{x \rightarrow a} \frac{ (3x+2)-(3a+2) }{ x-a }\]

Answer 4

Is that it?

Answer 5

simplify

Answer 6

\(\large \lim \limits_{x \rightarrow a} \frac{ (3x+2)-(3a+2) }{ x-a }\)

\(\large \lim \limits_{x \rightarrow a} \frac{ 3x+2-3a-2 }{ x-a }\)

Answer 7

\(\large \lim \limits_{x \rightarrow a} \frac{ (3x+2)-(3a+2) }{ x-a }\)

\(\large \lim \limits_{x \rightarrow a} \frac{ 3x+2-3a-2 }{ x-a }\)

\(\large \lim \limits_{x \rightarrow a} \frac{ 3x-3a }{ x-a }\)

Answer 8

\(\large \lim \limits_{x \rightarrow a} \frac{ (3x+2)-(3a+2) }{ x-a }\)

\(\large \lim \limits_{x \rightarrow a} \frac{ 3x+2-3a-2 }{ x-a }\)

\(\large \lim \limits_{x \rightarrow a} \frac{ 3(x-a) }{ x-a }\)

Answer 9

\[f'(x) = \lim_{h \rightarrow 0} \frac{ f(x+h)-f(x) }{ h }\]
\[f'(x) = \lim_{h \rightarrow 0} \frac{ (3(x+h)+2)-(3x+2) }{ h }\]
\[f'(x) = \lim_{h \rightarrow 0} \frac{ 3x+3h-3x }{ h }\]
\[f'(x) = \lim_{h \rightarrow 0} \frac{ 3h }{ h }\]
\[f'(x) = \lim_{h \rightarrow 0} 3\]
\[f'(x) = 3\]

Answer 10

\(\large \lim \limits_{x \rightarrow a} \frac{ (3x+2)-(3a+2) }{ x-a }\)

\(\large \lim \limits_{x \rightarrow a} \frac{ 3x+2-3a-2 }{ x-a }\)

\(\large \lim \limits_{x \rightarrow a} \frac{ 3(x-a) }{ x-a }\)

\(\large \lim \limits_{x \rightarrow a}~~~ 3\)

Answer 11

now that bottom cannot equal 0, you can substitute x = a

\(\large \lim \limits_{x \rightarrow a} \frac{ (3x+2)-(3a+2) }{ x-a }\)

\(\large \lim \limits_{x \rightarrow a} \frac{ 3x+2-3a-2 }{ x-a }\)

\(\large \lim \limits_{x \rightarrow a} \frac{ 3(x-a) }{ x-a }\)

\(\large \lim \limits_{x \rightarrow a}~~~ 3\)

\(\large 3\)

Answer 12

Also, you canfirm that 3 is correct answer by seeing that given funciton is simply a straight line wid slope of 3 :-

f(x)=3x+2
^