help beanie basic stuff calc

[woeful glen]

np

[tough summit]

oh

alr

[inner sand]

@woeful glen dms

[tough summit]

brb

back'\

back

u there?

@woeful glen r u speaking

[woeful glen]

$\derivative x f(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$

[regal basalt]

wolfqz

[tough summit]

cant see the stream btw

idk if its cuz of my shit wifi

or

alr

alr

[woeful glen]

formal definition

[tough summit]

secant line means graph of sec(x)

right

oh

[woeful glen]

straight line that intersects the curve at two pts

[tough summit]

yeah and what was h again

like

the displacement

or

the

curved

alr

yeah'

thats integration

right

oh

okay

[woeful glen]

$\derivative x x^2$

[regal basalt]

wolfqz

[woeful glen]

this

[tough summit]

we just subtitute h = 0

right

right.

mb

yeah

[woeful glen]

$\lim_{h\to 0}\frac{(x+h)^2 - x^2}{h}$

$\lim_{h\to 0} \frac{x^2+2xh+h^2-x^2}{h}$

[tough summit]

h+2x?

oh and how do we find lim of that

[woeful glen]

just plug h=0

as no more h in denominator

[tough summit]

yeah alr so the final ans

is 2x

right

[woeful glen]

Yeah

[tough summit]

alr

[woeful glen]

that is the derivative of x^2

[tough summit]

alr

[woeful glen]

now derivative at a point

x=a

2a

[tough summit]

okay

[woeful glen]

$\derivative x (ax^n+bx^{n-1}) = \derivative x ax^n + \derivative x bx^{n-1}$

[regal basalt]

wolfqz

[tough summit]

alright

[woeful glen]

$\derivative x (x^2+2x+1)$ at $x=5$

[regal basalt]

wolfqz

[tough summit]

12

damn i know calc

now

big boy stuff fr

[woeful glen]

yeah!!

real

[last sigil]

Shut the fuck up

[tough summit]

how tf do u create questions

arent u in 10th

oh

i thoguht u were

[woeful glen]

$\derivative x \sin x$

[regal basalt]

wolfqz

wolfqz

[tough summit]

thanks

no clue man

[woeful glen]

$\lim_{h \to 0} \frac{\sin(x+h)-\sin(x)}{h}$

[regal basalt]

wolfqz

wolfqz

[tough summit]

yeah

i got till

here

[woeful glen]

now

we do something tricky stuff

$\lim_{h \to 0} \frac{\sin(x)\cos(h) - \sin(x) +\sin(h)\cos(x)}{h}$

[regal basalt]

wolfqz

[tough summit]

u take it common?

i did that

[woeful glen]

$\lim_{h \to 0} \frac{-\sin(x)(1-\cos(h))}{h} + \frac{\sin(h)\cos(x)}{h}$

[tough summit]

wait

isnt it

cos(h) -1

[regal basalt]

wolfqz

[tough summit]

oh

[woeful glen]

$1-\cos(h) = 2\sin^2\frac{h}{2}$

[regal basalt]

wolfqz

[woeful glen]

$\cosaddition$

[regal basalt]

wolfqz

[woeful glen]

$\cos^2x - \sin^2x = 1-\sin^2x-\sin^2x = 1-2\sin^2x = \cos(2x)$

[regal basalt]

wolfqz

[woeful glen]

$x = \frac{h}{2}$

$1-2\sin^2 \frac{h}{2} = \cos h$

[regal basalt]

wolfqz

[woeful glen]

^

[woeful glen]

$\lim_{h \to 0} \frac{-2\sin(x) \sin^2\frac{h}{2}}{h} + \frac{\sin(h)\cos(x)}{h}$

$\lim_{h \to 0} \frac{-2\sin(x) \sin^2\frac{h}{2}}{h} + \lim_{h \to 0} \frac{\sin(h)\cos(x)}{h}$

$-2\sin(x) \lim_{h \to 0} \frac{\sin^2 \frac{h}{2}}{h}+ \cos(x)\lim_{h \to 0} \frac{\sin(h)}{h}$

[regal basalt]

wolfqz

[woeful glen]

$g(x) \leq f(x) \leq h(x)$\*

$\lim_{x \to a} g(x) = \lim_{x \to a} h(x) =L$

then $\lim_{x \to a} f(x) = L$

[regal basalt]

wolfqz

[woeful glen]

$\cos^2x \leq \frac{\sin x}{x} \leq 1$

[regal basalt]

wolfqz

[woeful glen]

^ can be proved using sectors and stuff

[tough summit]

alr

[woeful glen]

$\lim_{x \to 0} \cos^2 x = 1$

[regal basalt]

wolfqz

[woeful glen]

$\lim_{x \to 0} 1 = 1$

[regal basalt]

wolfqz

[tough summit]

so that limit is also 1

sin x/x

[woeful glen]

Yeah

[tough summit]

okay

[woeful glen]

back here

$-2\sin(x) \lim_{h \to 0} \frac{\sin^2 \frac h2}{h} + \cos(x)$

[regal basalt]

wolfqz

wolfqz

wolfqz

[woeful glen]

$-\sin(x) \lim_{h \to 0} \frac{\sin \frac h2}{\frac h2} \times \sin(0) + \cos(x)$

[regal basalt]

wolfqz

[woeful glen]

$0+\cos(x) = \cos(x)$