differentiation

[elfin iron]

Hint: use the identity sin²(√x) + cos²(√x) = 1 -> -sin(√x) = -√(1 - cos²(√x))

Here’s some steps to get you started:

cos(√x) = y
-sin(√x) * (1/2)(x^-(1/2)) = dy/dx

[stuck scaffold]

i dont really get the last line

[elfin iron]

The last line is just the derivative of cos(√x)

I take it that you do not know how to differentiate this?

[stuck scaffold]

ohhh nvm i see it now

[upper fable]

@stuck scaffold I don’t want to bother you, but if you get any time could you explain how you work it out

I don’t understand how you would start it

[violet harbor]

root both sides take cos function on both sides so u. can get rid of the arc cos on the RHS

so then u cos(x^1/2) = y

now differntiate wd respect to y

[upper fable]

I’m confused

[violet harbor]

cos and arccos are inverse fucntion so they cancel out each other

just ln and e

[upper fable]

wait so

For this question

Do you want to work backwards from the answer given

[violet harbor]

nah

[violet harbor]

idt we can do tht bcuz we would have the intergation costant + c if we integrate dy/dx

[upper fable]

I just don’t get how you would start the question

[stuck scaffold]

@upper fable square root both sides
rtx = arccos y
then cos it
cos rtx = y
differentiate using chain rule

[upper fable]

Arc cos y ^2

[stuck scaffold]

-sin(√x) = -√(1 - cos²(√x)) use identiity

[stuck scaffold]

dont expand it

[upper fable]

Ok I get it all up to cos root x = y

[stuck scaffold]

okay now how do we differentiate it

[upper fable]

I don’t know

[upper fable]

What is this identity

[stuck scaffold]

cos(rootx)

does this help?

[stuck scaffold]

sin^2 + cos^2 = 1
but theres a root x
so itll be sin^2 rt x etx
then rearrange to get sin the subject

[upper fable]

So rewrite as cos(x)^1/2 and then do chain rule

[stuck scaffold]

cos (x^1/2)

then yes

[upper fable]

Why in bradkets

[upper fable]

So sin^2 root x + cos^2 root x = 1

[stuck scaffold]

yes

[stuck scaffold]

when you rooted it all, you got rootx = arccosy

[upper fable]

what

[stuck scaffold]

get a piece of paper

and follow along

[upper fable]

I’m away from my paper

[stuck scaffold]

well its going to be hard to understand it unless you write it down

write it down anywehre

[upper fable]

I get it all up to cos root x = y

Could you send your working

[violet harbor]

to solve this now

u need to know how to differentiate a cos function

[upper fable]

yeah it becomes -sinx

[violet harbor]

what if u have cos(2x)

[upper fable]

Er

ok

So

Chain rule

?

@violet harbor

[violet harbor]

what u do is

[upper fable]

you don’t have a power

So it can’t be chain rule

[violet harbor]

so wht u do is

differntiate the input

the input in 2s

2x*

[upper fable]

yeah

[violet harbor]

u differnetiate tht u get 2

[upper fable]

So 2

[violet harbor]

now u multiply 2 by -sin(2x)

basically chain rule

[upper fable]

But there’s no power

[violet harbor]

when do we use chain rule

[upper fable]

plus we already done part of the chain rule here

[violet harbor]

when we have a funciton inside of a cuntion

[upper fable]

yeah

[violet harbor]

one function is 2x

which is inside the cos fucntion

[upper fable]

so do I imagine a power is there

[upper fable]

So dy/dx of this

[violet harbor]

power where

yeah so u hv

y = cos(x^1/2)

[upper fable]

I would rewrite it as cos(x^1/2)

yeah

[violet harbor]

differntiate the input

u get 1/2x^-1/2

[upper fable]

yeah

[stuck scaffold]

you forgot the sin

[violet harbor]

now multiply this by sin(x^1/2)

yeah was getting there

[stuck scaffold]

oh

[upper fable]

What rule is this??

[stuck scaffold]

thats a weird way of doing it lmao

[violet harbor]

lol

[upper fable]

I’m actually confused

[stuck scaffold]

chain

[upper fable]

I always have done chain rule with powers

[violet harbor]

i diff the input and just multiply by sin(original thingy)

[stuck scaffold]

cos(rx)
-sin(rx) x 1/2x^-1/2

[upper fable]

Ok I get that

[stuck scaffold]

okay nice

then you use the identitiy

[violet harbor]

to differetiate cos this is the formula: cos(f(x)) = f'(x).-sin(f(x))

[violet harbor]

this might be abit confusing tho ig

[stuck scaffold]

it does seem like you overcomplicated it 😭

[violet harbor]

lol

[upper fable]

How can we use an identity😭

Because I always thought sin^2 + cos^2 = 1 but you can’t do sinx+cosx=1 so you can’t rewrite the identity

If that makes any sense

[stuck scaffold]

sin^2 = 1 - cos^2
so sin = root (1-cos^2)

[elfin iron]

What’s the problem?

[elfin iron]

You would work backwards

Just do lots of these questions and you will get the hang of it!

[upper fable]

I just don’t get it

Im up to here cos(√x) = y

Now I differentiate

So cos(x^1/2)

So 1/2x^-1/2 x -sin root x

[stuck scaffold]

you have the formula sin^2 + cos^2 = 1

rearrange to get sin

[upper fable]

So sin^2 = 1-cos^2

[upper fable]

So sin^ root x = 1-cos^2 root x?

[stuck scaffold]

you still have sin^2

[upper fable]

what

This is where I get confused

[stuck scaffold]

how do you get rid of sin^2

[upper fable]

root it

[stuck scaffold]

yes

so root the entire expression

[upper fable]

But I didn’t think you could do that with that expression

Becuase that’s like saying sinx+ cos x = 1

[stuck scaffold]

it isnt

cause if you rearrange that expression (the one i told you) you will get back to sin^2 + cos^2 = 1

[upper fable]

So your saying sin^1/2 x = 1- cos^1/2 x

[stuck scaffold]

to the power of 1/2??

[upper fable]

idk

Oh wait

I’m lost

[stuck scaffold]

bro get some paper

[upper fable]

This shouldn’t be difficult tho

sin^2 + cos^2 = 1

[stuck scaffold]

youre confusing yourself, if you have paper you will have it stored and you will understand it better

[upper fable]

We need -sin root x

[elfin iron]

Yes

Subtract cos^2 from both sides

[upper fable]

Then that would leave sin^2 = 1 -cos^2

But we need -sin

[elfin iron]

Root both sides

Then multiply both sides by -1

[upper fable]

so sin= 1-cos

[stuck scaffold]

bro pls

[upper fable]

Oh wait

[stuck scaffold]

just get some paper

[upper fable]

lol

sin^3/2 = 1-cos^3/2

yeah?

[stuck scaffold]

bro i give up

[upper fable]

Same my grade is in the mud

[stuck scaffold]

WHERE HAS THE POWER OF 3/2 COME FROM MAN 😭 😭

[upper fable]

how do you root

Both sides

Where am I going wrong

[stuck scaffold]

bro pls just get some paper

1 sheet of paper is all you need

[elfin iron]

sin²(√x) + cos²(√x) = 1
sin²(√x) = 1 - cos²(√x)
sin(√x) = √(1 - cos²(√x))
-sin(√x) = -√(1 - cos²(√x))

What’s wrong with doing this?

[upper fable]

Oh my god

You are so right