Exponential derivative visual

A visual of the derivative of f(x)=e^x. We show how to think about the derivative of a function visually.

#manim #calculus #derivatives #derivative #tangentline #slope #parabola #mathvideo #mathshorts #math #visualmath #graph #exponential #linearapproximation

[dageustice]

f(x)=0 also has itself as its derivative, but no one seems to care at all ☹️

[sathmika]

we should use digital interfaces to teach maths to students. anyway your channel is awesome....

[lillii]

By the definition of derivative, we find that (a^x)' = (a^x)(a^h - 1)/h when h -> 0,
Let e be the number so that (e^h - 1)/h = 1,
e^h - 1 = h <=> e^h = 1 + h <=> e = (1+h)^(1/h)
By definition, this is the only real number that, as the base of an exponential function, gives that function itself as a derivative

[kitspapp]

i remember my maths professor showed us this proof in intro calc, my mind was blown

[jesusnoagervasini]

f(x)=0: Look at what they have to do to mimic a fraction of my power

[elnetini]

The next step would be to show the area under the curve, which is the same value as the slope at any given point

[gilly]

I hate my Calc 1 class rn but this video genuinely helped me. Thank you.

[loohooi]

In addition, the difinite integaral(the area below the curve) from negative infinity to any x value of that point is also exactly to its y-coordinates.

[skyzm]

Another intereating method is using taylor's expansion series to prove that for all integer n since e^x=1+x+x^2/2+...x^n/n .then it's derivative would remain the same neglecting the rest since we're doing a derivative

[pagoluharshavardhangowdme]

It also has another interesting property. Although it's slope increases exponentially it's tangents intercept on y axis increases linearly with x i.e the intercept of tangent at x=1 is 0, at x=2 is 1, at x=3 is 2.... And so on

[Umlaut]

Thank you! This helps with comprehending calculus

[pelayomedina]

That's literally how e^x is defined

[logosking]

Not by coincidence, but by definition

[Tiggster-qrmw]

Multiply e^x by any constant and the same thing still applies

[korakatk]

How did we find e in the first place? Pi and e are so famous but school never taught me the history behind e

[nikolatesla]

Sir please make a video on LEFT HAND DERIVATIVE and RIGHT HAND DERIVATIVE.
in the same way.

[runningonempty]

It is quite easy to understand once it's visualused like this, but personally I really like the other explanation of its consistency which involves Taylor series.
You see, e^x can be expressed as an infinite sum of x^n/n! so it goes like this: 1+x + x²/2!... and so on.... So if you find a derivative of this you'll get the exact same thing, and I really *love it*

[grimanium]

not only that, that also means the area under the curve is equal to the curvature which is equal to y :)

[mananagrawal]

I mean, that's how e is defined.

[georgecop]

we had to find out the derivatives of e^x, sin and cos using fundamental limit and the definition of the derivative. This is more easier, though it is less rigorous.