where did +x^2 in the second line come from?
#confusion about chain rule
lavish trellis
empty spireBOT
lavish trellis
do i have to use product rule or is this just chain rule
thick idol
do i have to use product rule or is this just chain rule
It’s the product rule
You have two functions, x^2 and tan(1/x), you take the derivative of the first and multiply it by the second, and add that to the first times the derivative of the second
lavish trellis
can i do this using chain rule or is this only possible using product rule? like im trying to figure out how they did it but idk
thick idol
can i do this using chain rule or is this only possible using product rule? like...
The chain rule would be applicable if you had d/dx( (tan(1/x)^2 ) because you have an f(x)=x^2 and g(x)=tan(1/x) in the form f(g(x))
Here there isn’t an f(g(x)), its f(x)g(x)
lavish trellis
im kind of confused which is which in the composite
is x^2=f(x) and then tan1/x = g(x)?
thats how i was trying to do it earlier
but if its product rule then nvm lol
thick idol
I’ll make it very explicit.
f(x)=x^2
g(x)=tan(1/x)
If we have f(g(x)), then it is equal to f(tan(1/x)) which is equal to tan(1/x)^2.
Consider our case in your problem, there it’s clear to see it’s in the form f(x)g(x) because the two functions don’t take one another as an argument, they’re distinct from each other
@lavish trellis
thick idol
Identify the composition by identifying the functional components, here there is no composition
lavish trellis
does there have to be a power for it to be a composite function
lavish trellis
like im struggling to identify whats composite and whats not
thick idol
tan(1/x) is itself a composite
So you first use the product rule, then the chain rule
lavish trellis
oh
thick idol
f(x)=tan(x)
g(x)=1/x
lavish trellis
when im using product i only do it for two functions right
thick idol
There’s only two functions
lavish trellis
and then for chain i would do tan and 1/x as a composite
thick idol
Yes
lavish trellis
lemme try something one sec
thick idol
Technically there’s 3 functions but in the context of the product rule there’s only two, it just so happens that the second function is a composite function so needs to be differentiated by the chain rule
So basically what u said
lavish trellis
ohhh
so when im doing product, i use chain rule for finding g prime of x if g(x) = tan(1/x)
but its sort of like that chain rule is within the product rule
lavish trellis
thank you
thick idol
No problem, you’ll run into problems like this a lot where you need to use two rules. Start looking broadly, then see what the broad rule requires you to do to solve for it (in this case it asks for g’(x) which requires the chain rule),
lavish trellis
i think another issue i have is when to use quotient rule or chain rule, because you can treat fractions like composites since its just x^-1 where x is the inner function and ^-1 is just the fraction itself
like should i be avoiding quotient rule entirely because of that or are there cases where i can only use quotient
heres an example
sry heres how the answer key did it because im like 90 percent sure i did that wrong lol
thick idol
The quotient rule is used when there's a function f(x) and g(x) in the form f(x)/g(x) and we're asked to differentiate. 1/x can itself be considered a function without composition
we're just taking our x and taking the reciprocal, which is one function
lavish trellis
so in this case i can use either quotient or chain and itll yield the same result?
thick idol
That function is multiplicative inversion (except for when x=0)
lavish trellis
like i dont know when i should be using quotient because in the example above i think they used chain rule but i could also use quotient rule
thick idol
so in this case i can use either quotient or chain and itll yield the same resul...
Let me check rq
Yes you could
lavish trellis
wait i think if the numerator were something other than one i wouldnt be able to use chain simply like they did
lavish trellis
because the numerator is one, you can treat it like a composite right? since the fraction is sort of like the denominator to the power of ^-1
thick idol
It wouldn't be a composite unless f(x)=(1/x) g(x)=x f(g(x))=1/x which requires us to take the derivative of f(x), which throws us into an infinite loop
lavish trellis
oh
thick idol
It might be right to say that f(x)=(1/x) g(x)=x f(g(x))=1/x is trivial, or even redundant
g(x) is already within f(x) so it's circular
lavish trellis
wdym by its circular
thick idol
If we tried to compose the two and take the derivative, we'll be in an infinite loop because there's no way to find the derivative without finding the derivative which can only be found by finding the derivative and so on
f(x)=(1/x) g(x)=x f(g(x))=1/x, by the chain rule f'(g(x))*g'(x), and to find f'(g(x)) we need to find the derivative of f(x) and we're thrown back into the loop when we compose them again to take the derivative
lavish trellis
im kind of confused because in the example above they did it using chain rule, so is it possible or not?
thick idol
Chain rule wouldn't apply to 1/x is what I'm saying
lavish trellis
doesnt chain rule apply to any function to the power of something
so like you could chain rule x^-1 where x = some other function
thick idol
The difference between what I'm saying and the problem you posted is that the bottom isn't just x
x^-1 is different than (t^2+blah blah blah+blah)^-1
lavish trellis
oh
yeah i think im confused about that because im also struggling to understand why sinx isnt a composite, but sin5x is
thick idol
x^-1, power rule -x^-2 = -1/x^2, which is the same answer, but there's no composite because the function itself is (x^-1) which means we don't need the chain rule
thick idol
yeah i think im confused about that because im also struggling to understand why...
Because the x is unmodified. A function is a transformation, f(x)=x maps x to itself, so it's trivial
5x however will scale x by 5
lavish trellis
like from what im getting rn if its just a basic function with only x, then its not composite, but if theres other constants or coefficient then it becomes a composite?
ohh right transformations
wait how do transformations relate to composite functions?
thick idol
Because functions are transformations, and we're trying to find the transformation of a transformation
or the function of a function, such as f(g(x))
If g(x) is just g(x)=x, we can just say f(x)
lavish trellis
ohhhh
thick idol
because g(x)=x
f(g(x))
Sub x for g(x)
f(x)
lavish trellis
so its redundant
lavish trellis
i see
thick idol
A function which maps something to itself is just the thing itself
lavish trellis
but as soon as you add a coefficient or something alongside the x, its actually worth creating that other function
thick idol
Of course, because 5x doesn't equal x, so if we have our example from before f(g(x)) where g(x)=5x, then f(5x) which is not f(x), so we can't just say f(x)
We only say x where we mean x=5x (as in the chain rule when we're identifying the outer function)
lavish trellis
are there ever cases where 5x would equal x? like for example if x is 0 what would happen, like are there restrictions when we work with composite functions or is it just assumed that x cant be 0?
thick idol
We can assume the whole domain and everythings fine, because the exception doesn't prove an exception for every number
If there were an exception for every number that would be the case, but there's only one number where that's the case
lavish trellis
oh true
thick idol
So we can't say x=5x in general, but only when x=0
lavish trellis
it only applies for that one number but we dont know that x is equal to only zero so
yeah
i see
thick idol
it only applies for that one number but we dont know that x is equal to only zer...
Exactly! And if we did, then you'd be right
But f(x) abstracts from any particular number, which is the point of the variable
It's only when we bring the abstract back into the concrete that we need to consider the exceptions (like when we say 1/x is defined except when x=0)
thick idol
Of course
lavish trellis
im gonna try some more of these questions and see if i have any trouble
i think i was just stuck in the idea that i could just do everything using only chain rule, i forgot i could also apply product and quotient to make my life easier as well lol
im wondering where they got -x^2 from in the second line?
the first img is the key
like im closer to the answer key than i was 30 minutes ago but i cant figure out where that negative came from
@thick idol if you're still free to look over this would be great
thick idol
<@148238632740847616> if you're still free to look over this would be great
Finishing a little homework then I'll take a look. Anyone else feel free to help in the meantime