Differentiation Flashcards
How to differentiate from first principles?
x^2 + 2x + 5 = f(x)
1) f(x) becomes f(x+h)
so now it’s from x^2 + 2x + 5
to
(x+h)^2 + 2(x+h) + 5 (changing only the x’s)
2) (f(x+h) - f(x))/h () = eh it’s all / by h
(x+h)^2 + 2(x+h) + 5 - x^2 + 2x + 5
course after working all that out,
final answer = 2x+h+2
3) dy/dx limit, h has gotta be 0
therefore, 2x+h+2
= 2x+2
u just remove all the h’s or the one’s involved with h
(e.g. 3x^2+3xh+h^2 = 3x^2)
How to differentiate not in first principles?
differentiate = f’(x)
The general rule issssss:
anx^n-1
e.g.
3x^2 = 6x or 3x^5 = 15x^4
What’s the first differentiation for?
1) finding gradient
y = 3x-8x^2 at p(3,-63)
dy/dx = 3-16x at x=3
dy/dx=3-48 = -45 <– M
2) Solving for x
y = x^3+5x+4 where gradient = 17
dy/dx = 3x^2+5
3x^2+5 = 17
solve for x = x is + or - 2 after rearranging type shit but also cus theres a root?
3) ^^^ additionally finding the points
sub x into y = x^3+5x+4
coordinates gained:
(2,22) (-2,-14)
4) Finding stationary pts
I mean same as 2)
5) rate of change
mehhh
How to tell if it’s an increasing function?
dy/dx > 0
How to tell if it’s a decreasing function?
dy/dx < 0
How to do second derivation?
Same as first derivation, do it again
What’s the second differentiation for?
Finding out if it’s a max/min value
How to tell if it’s maximum value?
d^2y/dx^2 < 0
How to tell if it’s minimum value?
d^2y/dx^2 > 0