Applying the binomial theorem to a fractional exponent seems a bit hand-wavy to me. Never fear, there is still a way to salvage the proof tho. First, start by proving the chain rule from first principles. Having done that the Inverse Function Theorem follows almost by inspection. Next, if y = x^(1/100) then y^100 = x. One can easily find the derivative of y^100 from first principles using the binomial theorem. Having done that apply the Inverse Function Theorem to this result and the derivative of x^(1/100) is thus proved.
This article is above my pay grade, but if I'm reading it right -- IF -- maybe not so many hands were waved. en.wikipedia.org/wiki/Binomial_coefficient#Generalization_and_connection_to_the_binomial_series
@@kingbeauregard Thanks for the link. That's good information to have, but still not something I would expect beginning calculus students to know. For what it's worth I'm sure Newton's explanation was correct; it's just the why is it correct part that was unclear to me.
Hi, teacher. Thanks for the amazing videos! I haven't tried this yet but couldn't one simply transform the exponent into a radical and then multiply by the conjugate?
The first principles formula, aka the difference quotient is the slope of the line tangent to the curve f(x): change in the y-coordinates over the change in the x-coordinates. You then evaluate said slope as the change in the x-coordinates goes to zero. The resulting limit is the definition of the derivative. That's why this particular limit is said to be from first principles.
Thank you but there is no end of the term and the last term is not h^(1/100) h is increasing by 1 each time and continue on without stop becase the exponent of x will not go to 0 so its exponent of h will never become 1/100
Very nice! Also, I think that's my favorite hat of yours. I would follow that hat to hell and back.
Thank you so much Sir excellent teaching.....
Prime Newtons is awesome! 😊
وشكرا استاذ طريقة ممتاز ة
Thank you very much
You're welcome 😊
The beat activating
Applying the binomial theorem to a fractional exponent seems a bit hand-wavy to me. Never fear, there is still a way to salvage the proof tho. First, start by proving the chain rule from first principles. Having done that the Inverse Function Theorem follows almost by inspection. Next, if y = x^(1/100) then y^100 = x. One can easily find the derivative of y^100 from first principles using the binomial theorem. Having done that apply the Inverse Function Theorem to this result and the derivative of x^(1/100) is thus proved.
This article is above my pay grade, but if I'm reading it right -- IF -- maybe not so many hands were waved.
en.wikipedia.org/wiki/Binomial_coefficient#Generalization_and_connection_to_the_binomial_series
@@kingbeauregard Thanks for the link. That's good information to have, but still not something I would expect beginning calculus students to know. For what it's worth I'm sure Newton's explanation was correct; it's just the why is it correct part that was unclear to me.
That makes a lot of sense
Hi, teacher. Thanks for the amazing videos!
I haven't tried this yet but couldn't one simply transform the exponent into a radical and then multiply by the conjugate?
thanks
As the power of h increases, shouldn't it be h^100 in stead of h^(1/100) for the last term?
how do you derive the formula for first principles?
The first principles formula, aka the difference quotient is the slope of the line tangent to the curve f(x): change in the y-coordinates over the change in the x-coordinates. You then evaluate said slope as the change in the x-coordinates goes to zero. The resulting limit is the definition of the derivative. That's why this particular limit is said to be from first principles.
That's the slope of the tangent line.
Thank you but there is no end of the term and the last term is not h^(1/100)
h is increasing by 1 each time and continue on without stop becase the exponent of x will not go to 0
so its exponent of h will never become 1/100