Question

differentiation without knowing or understanding limits, is this even possible even without knowing the rules?

Answer 1

Uhh...
Cheat?
Just take to faith the power rule, the chain rule, and what-not... maybe?

Answer 2

you can differentiate simple functions sure.

Answer 3

without knowing the power rule, chain rule, product rule, quotient rule...

Answer 4

Even then, it'll be extremely difficult to appreciate, say, the mean value theorem, or other theorems that involve that immortal form:
lim h-->0 [f(x+h) - f(x)]/h

Answer 5

you could just use the limit definition without limit notation and just plug crap in

Answer 6

you just used limit right there

Answer 7

i know. but if i just gave you f(x+h)-f(x)/h
without the limit notation. you could still find the derivative.

Answer 8

Well in that case, I doubt it, since, well, the derivative (slope of the tangent line?) relies on the tangent line itself being a secant line where the two points it intersects get really really close to each other (pretty much being a tangent line)

Answer 9

why would one begin to use f(x+h)-f(x)/h to differentiate?
what is the idea behind this expression?

Answer 10

some of us need to untaught and be fool again in that case.... but there has to be another method

Answer 11

the limit definition. or you could call h delta x if you want. since i learned derivatives independent of class. i learned derivatives this way first. with just regular polynomials.
and i didnt understand limits

Answer 12

Slope of a function is (change in y)/(change in x) ... (would look much nicer in TeX)

Let's call the first point in the function (x1, y1), and the second point (x2, y2).

That makes the slope equal to
(y2 - y1)/(x2 - x1)

If y can be written as a function in terms of x, then you can say the slope is equivalent to
( f(x2) - f(x1) ) / (x2 - x1)

Let's suppose we want to write x2 in terms of the first point. Let's also call the difference between the two values of x, h. So that means we're setting
x2 - x1 = h
Or, equivalently, x2 = x1 + h.

So now the slope equation gives us
( f(x1 + h) - f(x1) ) / (x1 + h - x1)

Note the simplification that occurs:
( f(x1 + h) - f(x1) ) / h

Now, to make it easier on they eyes, let's just call x1, x, so that we have
( f(x+h) - f(x) ) / h.

We call this the difference quotient. If you can wrap your mind around the limit concept, then you'll immediately see why the derivative is the instantaneous rate of change.

Answer 13

I agree.

Answer 14

I hope the wall of text wasn't too intimidating...

Answer 15

I am not intimidated at all