if f(x) = x^n
then d f(x)/dx = n x^(n-1)
if f(x) = x^2
then d f(x)/dx = 2 x
so if f(t) = t^2
df(t)/dt = 2 t
the constant multiple remains (three times the function has three times the slope) .
if f(t) = 2.1 t^2
d f(t)/dt = 4.2 t
Now about that constant B
The slope of a constant is zero.
d/dt (A t^2 + B)
= 2 A t + 0
the end except to prove that d/dx (x^n) = n x^(n-1)
f(x+dx) = (x+dx)^n
(x+dx)^n = x^n + C(n,1) x^(n-1) dx + C(n,2) x^(n-2) (dx)^2 ....... +dx^n
C(n,k) = binomial coefficient
= n!/[(n-k!) k!]
so for Cn,1)
C(n,1) = n!/[(n-1)!] = n (lo and behold)
f(x+dx) = x^n + n x^(n-1)dx + terms in dx^2 through dx^n
f(x+dx) -f(x) = n x^(n-1) dx + terms in dx^2 to dx^n
df(x)/dx = [f(x+dx)-f(x)]/dx
as dx ---> 0
= n x^(n-1)
f(x) = x^n
Ignore the last two lines, I was thinking out loud.
You may not need that proof at the end that d x^n/dx = n x^(n-1) but it is not a bad idea to know where the derivative comes from.
Math - In solving the eqaution(x+4)(x-4),Eric stated thatthe solution would be x...
math 10 slopes eqaution!!! HELP! please - solve and comparison: a)4x-9y+4 6x+15y...
Physics please help - Geosynchronous satellites orbit the Earth at an altitude ...
AP Chemistry - Consider ammonia, what is the pH of a buffer solution prepared by...
physics - please explain further - Can someone please explain the answer I got ...
AP Calculus - High School - The Temperature T of food put in a freezer is T = (...
Physics - PLEASE HELP!! The x and y cooridinates of an object are given by x(t...
Math - Can you please explain this to me. The problem is 7/16 - 3/20 First I ...
Physics repost please check - A glass bottle of soda is sealed with a screw cap...
AP CHEM - A secondary alcohol results from the attachment of a hydroxy group to ...