find [dy/dx] x^2
Using this :
limit as the change of x —-> 0 for
f(x + deltax) - f(x)
—————————-
deltax
run through each step
f(x+dx) = (x+dx)^2
= x^2 + 2 x dx + dx^2
f(x) = x^2
------------------------ subtract
f(x+dx) - f(x) = 2 x dx + dx^2
divide by dx
[ f(x+dx) - f(x) ] / dx = 2 x + dx
let dx ----> 0
= 2 x
the end
d/dx (x^2) = 2 x is now established in your mind forever.
now if f(x) = x^n
f(x+dx) = (x+dx)^n = x^n + n x^(n-1) dx + [n(n-1)/2!]x^(n-2) dx*2 ..... + ....dx^bigger
f(x) = x^n still
-------------------------subtract
f(x+dx) -f(x) = n x^(n-1)dx+[n(n-1)/2!]x^(n-2) dx*2 .....
divide by dx
[f(x+dx) -f(x)] /dx = n x^(n-1) + [n(n-1)/2!]x^(n-2) dx ..... +...dx^ bigger
let dx ---> 0
d/dx(x^n) = n x^(n-1)
so for n = 5173
d/dx(x^5173) = 5173 x^5172 :)
To find the derivative of a function using the limit definition of derivative, follow these steps for the function f(x) = x^2:
Step 1: Start with the limit expression:
lim (deltax -> 0) [(f(x + deltax) - f(x)) / deltax]
Step 2: Substitute the function f(x) = x^2 into the expression:
lim (deltax -> 0) [((x + deltax)^2 - x^2) / deltax]
Step 3: Expanding the numerator:
lim (deltax -> 0) [(x^2 + 2x * deltax + (deltax)^2 - x^2) / deltax]
Step 4: Simplifying the numerator:
lim (deltax -> 0) [(2x * deltax + (deltax)^2) / deltax]
Step 5: Cancel out the deltax term in numerator and denominator:
lim (deltax -> 0) [2x + deltax]
Step 6: Take the limit as deltax approaches zero:
As deltax approaches zero, 2x + deltax approaches 2x.
Therefore, the final derivative of f(x) = x^2 is:
dy/dx = 2x