Yes, this is a good video. Teaching math is not about solving the entire problem and taking long time to simplify. It should be about teaching patterns, techniques and showing applications of rules. I teach my son by showing him 100 problems and he has to talk to me to tell me what he sees on each problem and what he plans to do. Not waste time multiplying decimals.
Given: e^(ln(sin(arcsin(x)))) Assign the function names, f(x), g(x), h(x), and p(x), to show the chain rule process: f(x) = e^x g(x) = ln(x) h(x) = sin(x) p(x) = arcsin(x) Corresponding derivatives: f'(x) = e^x g'(x) = 1/x h'(x) = cos(x) p'(x) = 1/sqrt(1 - x^2) First chain rule: d/dx h(p(x)) = h'(p(x)) * p'(x) Next chain rule: d/dx g(h(p(x))) = g'(h(p(x))) * h'(p(x)) * p'(x) Final chain rule: d/dx f(g(h(p(x)))) = f'(g(h(p(x)))) * g'(h(p(x))) * h'(p(x)) * p'(x) Thus: d/dx e^(ln(sin(arcsin(x)))) = e^(ln(sin(arcsin(x)))) * 1/(sin(arcsin(x))) * cos(arcsin(x)) 1/sqrt(1 - x^2) As an exercise to you, this will simplify to 1. Which we expect, because your given function is all compositions of inverses, that ultimately simplifies to x.
Yes, this is a good video.
Teaching math is not about solving the entire problem and taking long time to simplify.
It should be about teaching patterns, techniques and showing applications of rules.
I teach my son by showing him 100 problems and he has to talk to me to tell me what he sees on each problem and what he plans to do.
Not waste time multiplying decimals.
Now I know all about The Box
This came just in time for my derivatives test so talk about awesome timing thank you!
1. e^(sqrt(x)) * 1/(2sqrtx) = e^sqrt(x)/(2sqrt(x))
2. sec(x^2 + x)tan(x^2 + x) * (2x + 1) = (2x + 1)sec(x^2 + x)tan(x^2 + x)
3. (cosx(1 + cosx) + sin^2x)/(1 + cosx)^2 = (cosx + 1)/(1 + cosx)^2 = 1/(1 + cosx)
4. 2xtanx + x^2sec^2x
Every time I hear you say box I suffer less, thank you
Always The Box.
It's like a mathematical horror movie.
The GOAT
I don't have access to Patreon from my countrey, could you please share the link via other clouds?
Some countries block Patreon? That's just evil.
Please take this👑
Is there any need to further simplify the answer ?
Wheres the derivative table?
That is also in the first Patreon link. You just need to be a free member to access it.
@@bprpcalculusbasics Thanks
So cool
Some are not simplified
Yay
Now differentiate e^ln(sin(arcsin(x))) no simplification
Given:
e^(ln(sin(arcsin(x))))
Assign the function names, f(x), g(x), h(x), and p(x), to show the chain rule process:
f(x) = e^x
g(x) = ln(x)
h(x) = sin(x)
p(x) = arcsin(x)
Corresponding derivatives:
f'(x) = e^x
g'(x) = 1/x
h'(x) = cos(x)
p'(x) = 1/sqrt(1 - x^2)
First chain rule:
d/dx h(p(x)) = h'(p(x)) * p'(x)
Next chain rule:
d/dx g(h(p(x))) = g'(h(p(x))) * h'(p(x)) * p'(x)
Final chain rule:
d/dx f(g(h(p(x)))) = f'(g(h(p(x)))) * g'(h(p(x))) * h'(p(x)) * p'(x)
Thus:
d/dx e^(ln(sin(arcsin(x)))) =
e^(ln(sin(arcsin(x)))) * 1/(sin(arcsin(x))) * cos(arcsin(x)) 1/sqrt(1 - x^2)
As an exercise to you, this will simplify to 1. Which we expect, because your given function is all compositions of inverses, that ultimately simplifies to x.
God revealed to me in a dream that the answer is 1.
@@carultch This literally feels like chatgpt write it 😭😭
FullSimplify[RSolveValue[(f[x] - f[x - a])/a == 1 - 2 x f[x], f[x], x] /. C[1] -> 0 /. a -> 1/z] as z->Infinity
What is the function?