Finding the Derivative: How to Solve for f'(x) Using the Limit Definition

[CrossFit415]

f(x)=3x+2 ; find [f(a+h)-f(a)] / h

So I got (3h+4)/h

But how did the book get 3? I tried everything. It must be an error.

 

[l-1j-cho]

Assuming that
[f(a+h)-f(a)] / h
was actually
lim(h->0) [f(a+h)-f(a)] / h
the expression above is the first principle formula to get a derivative of a function
therefore, f'(x) = 3
Please don't rely on my solution entirely and please refer to others'

by the way, I am just curious that, how did you get (3h+4)/h?

 

[I like Serena]

f(x)=3x+2 ; find [f(a+h)-f(a)] / h

So I got (3h+4)/h

But how did the book get 3? I tried everything. It must be an error.

How did you start?
When I calculate it, I get 3.

 

[AwesomeSN]

I guess you didn't learn the differentiation rules yet? Or do you have to find the answer using that formula?

But yes, the answer would be 3.

 

[CrossFit415]

Well...

[3(a+h)+2-3(a)+2] / h
=(3a+3h+2-3a+2) / h
=(3h+4) / h

 

[l-1j-cho]

conventionally, the following expression
lim(h->0) [f(a+h)-f(a)] / h
means the slope of the function at point a.

by the way, that's a creative thinking. Nice attempt.

In that expression
a is a certain 'point' at the function f(x) and h is a certain 'distance' from the point a.
You might already know that slope of a function is defined as
delta y / delta x
so let the slope between two points, a (a, f(a)) and p (a+h, f(a+h))
then the slope is defined as
(f(a+h)-f(a)) / (a+h - a)
which is [f(a+h)-f(a)] / h
now as h approaches to 0, you will get an instaneous slope of the function at the point a

 

[I like Serena]

Well...

[3(a+h)+2-3(a)+2] / h
=(3a+3h+2-3a+2) / h
=(3h+4) / h

I see, you have left out a couple of parentheses, effectively mixing up additions and subtractions.

Try it with the following:

[(3(a+h)+2) - (3(a)+2)] / h

Cheers!

 

[AwesomeSN]

Yes^^

 

[micromass]

Well...

[3(a+h)+2-3(a)+2] / h
=(3a+3h+2-3a+2) / h
=(3h+4) / h

Your mistake is in the first line: f(a)=3a+2, thus -f(a)=-3a-2...

 

[CrossFit415]

I see, you have left out a couple of parentheses, effectively mixing up additions and subtractions.

Try it with the following:

[(3(a+h)+2) - (3(a)+2)] / h

Cheers!

Ahh careless mistakes again! Thank you!

 

[l-1j-cho]

so it turns out to be that it is not a differntiation problem?

 

[AwesomeSN]

It is a differentiation problem.

 

[micromass]

Nothing tells us that it's a differentiation problem... He never took the limit of anything. For all we know, this could be an exercise in calculating with letters...

 

[AwesomeSN]

I see..I was just under the impression that h approaches 0, and the derivative of that function is 3.

 

[I like Serena]

This would typically be the set up in a math textbook just before introducing the derivative.
It starts with explaining slopes of functions over an interval h.

In this specific example the limit does not have to be taken to get the same result a derivative has, because the derivative is a constant function.