/sci/ - Science & Math

What specifically are you confused about? The many ways to find one, or what they actually are?

delta Y over delta X
:D

instantaneous slope of the line is it's derivative.
now fuck off.

There's a lot we could say about them. It would help if you give us a point to start at.

The derivative is the slope of the tangent line at any point of the function.

Working on about as thorough of an explanation I can provide OP, give me a few more minutes.

a derivative of a function is that function's slope. for instance, y=2x has a constant slope of 2, and sure enough, the derivative of 2x is 2.

>Not understanding what a derivative is.

ITT: some kid is fucked for his 1st semester calc final

Delivard

A derivative is the instantaneous rate of change at some point on a function. It's basically slope, but for curves.

The equation for the slope of a line is <span class="math"> \displaystyle{ \frac{ \Delta y }{ \Delta x } = \frac{y_2 - y_1}{x_2 - x_1} } [/spoiler]. This can be rewritten in function notation as <span class="math"> \displaystyle{ \frac{f(x_2) - f(x_1)}{x_2 - x_1}} [/spoiler]. These lines are called secant lines when our function is a curve because the line will cross the curve at least twice. It is the average rate of change of the function over the interval <span class="math"> [x_1 , x_2 ] [/spoiler]

The point of the derivative is to turn this secant line into a tangent line, which will be the instantaneous rate of change at that point. When we used function notation, we used to different x values: <span class="math"> x_1 [/spoiler] and <span class="math"> x_2 [/spoiler]. If we subtract them (<span class="math"> x_2 - x_1 [/spoiler]), we will get a constant, usually called <span class="math"> h [/spoiler] or <span class="math"> \Delta x [/spoiler]. If we add h and one of our x values, we'll get the other x value (<span class="math"> x_1 + h = x_2 [/spoiler]). 

Knowing this, we can rewrite our slope formula as <span class="math"> \displaystyle{ \frac{f(x+h) - f(x)}{(x+h)-x} = \frac{f(x+h) - f(x)}{h} } [/spoiler]. This is the average rate of change of change of the function over the interval <span class="math"> [x,x+h] [/spoiler]; it is still a secant line. In order to make this a tangent line, we need to make h equal zero. However, we can't divide by zero, so, instead, we observe what happens as h approaches zero, ie. we take the limit of the formula as h approaches zero (<span class="math"> \displaystyle{ \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} } [/spoiler]). This is the limit definition of the derivative. 

>>3027349
That's all calc 1 is about? I'm pretty sure I aced my AP Calc exam but I'm choosing to not take the credit if I actually did, so I can get into the swing of college on an easy note.

>>3027392 continued

I'll do an example to show you how it works. Let's say I want the derivative of <span class="math"> f(x)=x^2 +1 [/spoiler] at the point x=2.

First, plug the function into your formula.

<span class="math"> \displaystyle{ \lim_{h \to 0} \frac{ \left[(x+h)^2 +1 \right] - (x^2+1)}{h} } [/spoiler]

Now, we'll expand the binomial and simplify the numerator.

<span class="math"> \displaystyle{ \lim_{h \to 0} \frac{ \left[(x+h)^2 +1 \right] - (x^2+1)}{h} = \lim_{h \to 0} \frac{ (x^2 +2xh+h^2 +1) - (x^2+1)}{h} =\lim_{h \to 0} \frac{ 2xh+h^2 }{h} } [/spoiler]

Now, we will cancel out the h in the denominator. This is how we "avoid" dividing by zero.

<span class="math"> \displaystyle{ \lim_{h \to 0} \frac{ 2xh+h^2 }{h} = \lim_{h \to 0} 2x+h } [/spoiler]

Now, just plug in 0 for h and you have the general derivative of <span class="math"> x^2+1 [/spoiler]. To find the slope at x=2, simply plug 2 in for x.

<span class="math"> f'(x) = 2x, ~ f'(2)=2(2)=4 [/spoiler] 

The slope of the tangent line of <span class="math"> f(x)=x^2+1 [/spoiler] at x=2 is 4.

>>3027398
Calc 1 is derivatives, a little integration, and an intro to differential equations. Calc 2 is more integration, derivatives and integrals of parametric and polar equations, convergent and divergent sums, Taylor and Maclaurin series, etc.

>>3027416
Thanks...is that stupid 3d washers and shit in Calc1? Easy as all fuck but I think its stupid, even though I'm sure it has its applications.

>>3027422
It was taught in my AP class and was on the AB exam, so I would assume that they are. I thought they were cool...

>>3027429
You take the AP Test last week? Fuck the landfill problem, only one I couldn't completely do. I knew how to solve the differential given the initial conditions, but couldn't do the other parts.

>>3027440
Really? I thought the landfill was pretty easy. The one with the quarter circles and line fucked me sideways, though.. I can't do those to save my life :(

>>3027454
That one gave you an integral and the graph right?

>>3027440
FUUUUUUUUUUUCK the landfill problem.

:|

>>3027465
Yeah, something like <span class="math"> g(x) = 2x + \int_{0}^{x} f(x) dx [/spoiler]

I always get fucked up when x is negative...

>>3027482
Just throw a negative in front of the integral and flip the upper and lower limit things.

>>3027493
I know that much, but when you had to differentiate to find the max, it fucked everything up :(

>>3027505
Just plug in the x as you derived it!

>>3027573
Yes, but then you have to do a sign chart. A max is a chain from + to -, and, because you change the sign, there was only a - to +.

There's a good chance I did it wrong and and am just talking out of my ass, though.. :/

>>3027392
>>3027399

Good job...I guess no one cares though lol

>>3027392
>>3027399

Beautiful.