# Differenciation
=> integration.gmi Also see Integration

Differenciation was discovered/pioneered by Isaac Newton
It allows you to find the gradient equation for a curve (eg x²+3x+6)
You can easily find the gradient between two points on a curve by using the equation

```
change in y / change in x
```

But if you want to find the gradient at a specific point on a graph, you have to draw a tangent... Or DO you?
Well, if you pick two points on a graph (f(x): y = x²+3x+6) where 
```x = 1```
and 
```x = 1 + h```
Where h is a number that is very close to 0.
You can find the gradient between these two points:

```
change in y
-----------
change in x

=>

((1+h)² + 3(1+h) + 6) - (1² + 3(1) + 6)
---------------------------------------
          (1 + h) - 1

=>

( h² + 5h + 10) - (10)
----------------------
           h

=>

h² + 5h
-------
   h

=>

h + 5

=>

5 (because h is nearly 0)
```

...thus the gradient at point x=1 is 5
This proof is called First Principle and can be written:
```
             f(x+h) - f(x)
             -------------
f'(x) = lim        h
        h→0
```
This takes a while so a quicker way we can do this is by differenciating the curve equation
We do this by multiplying the coefficient by the number that the x is a power to and then minusing one from the the power:

```
f(x) = x² + 3x + 6
f'(x) = 2x + 3
dy
--  = 2x + 3
dx
```

f'(x) and dy/dx are just different ways of denoting a differenciated equation
Then we can simply substitute 1 into the equation:

```
2(1) + 3 = 5
```

And boom, that is differenciation; finding the gradient line of a curve
If you draw the gradient curve then you will see that the point it crosses the x axis, where x=0, the gradient=0 as that is the turning point of the quadratic equation:

```
0 = 2x + 3
-3 = 2x
-1.5 = x
```

And so x = -1.5 is the minimum point for x

```
x² + 3x + 6:
       |       | |
        |      ||
        |      ||   (1. Sides of the V shape are getting steeper)
        |      ||
         |     |
          '   '|
           \./ | (2. Turning point is -1.5)
_______________|________________
               |
               |
2x + 3:
               |   /
               |  /
               | /
 (2. Crosses   |/    (1. Because the line is getting higher)
 the x axis    /
  at -1.5)    /|
             / |
____________/__|________________
           /   |
          /    |
```
Additionally, if an equation has roots or fractions, you can use the index laws to put the equation in indicies form:
=> index-laws.gmi Index Laws
```
x³ + sqrt(x)
= x³ + x^(1/2)

2x² + 1/x
= 2x² + x^(-1)
```

Questions:
1.
a) Differenciate x³ - 4x² + 6x - 1
b) Find the gradient at x=3

=> integration.gmi Also see Integration
for finding the area under a curve, the reverse process of differenciation

Answers
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Answers:
1.
a) 3x² - 8x + 6
b) 9