Differentiation

Question 1

How to differentiate from first principles?

x^2 + 2x + 5 = f(x)

Answer 1

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)

Question 2

How to differentiate not in first principles?

Answer 2

differentiate = f’(x)

The general rule issssss:
anx^n-1
e.g.
3x^2 = 6x or 3x^5 = 15x^4

Question 3

What’s the first differentiation for?

Answer 3

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

Question 4

How to tell if it’s an increasing function?

Answer 4

dy/dx > 0

Question 5

How to tell if it’s a decreasing function?

Answer 5

dy/dx < 0

Question 6

How to do second derivation?

Answer 6

Same as first derivation, do it again

Question 7

What’s the second differentiation for?

Answer 7

Finding out if it’s a max/min value

Question 8

How to tell if it’s maximum value?

Answer 8

d^2y/dx^2 < 0

Question 9

How to tell if it’s minimum value?

Answer 9

d^2y/dx^2 > 0