Indices And Differentiation

Yesterday, courtesy of my beautiful and awesomely intelligent maths tutor, I suddenly worked out a way of making differentiation with negative indices slightly easier.

When inputting the x value to find the gradient at that point, to make the final stage of adding it all up much easier, turn your x^-2 into 1/x². It makes calculations a lot, lot easier.

Laws of Indices

Are pretty simple.

a^m x aⁿ = a^m+n

a^m / aⁿ = a ^m-n

(a^m)ⁿ = a ^mxn


But a^m + aⁿ DOES NOT equal anything other than what it says.
There isn't much you can do to collect the terms if the indices are different, or if the constant is different (ie x^m + y^m)

a^m + a^m = 2(a^m), of course.

Fractional Indices (2)

Well yes it gets a bit harder.

You see...a ^n/m


means ^m√aⁿ


...but I will try to explain this later.

Fractional Indices (1)

Think about it.


Think about it.


x is not just x. x is x¹. x is x raised to the power of one: x multiplied by itself no times at all; x just being x. So then x ¹ is x on its own. x is, therefore, x.

So then, x^1/2 is going to be a number that, multiplied by itself, makes x.

In other words, the square root of x.


√x = x ^1/2


It does get a bit more complicated than this....

Negative Indices

Negative Indices

Now these seem simple enough: you raise a number to a power and it usually means multiplying a number by itself n times.

a³ = a x a x a

All numbers to the power of 1 are themselves, and all numbers to the power 0 are one.

a⁰ = 1



Easy enough.


But then you can also have negative powers.

a^-3

What? You can’t multiply a number by itself a negative number of times!

Well, no, clearly. But you can see how negative powers come about.


It’s to do with the powers rule.

aⁿ x a^m = a^n+m

You can see this if you write out aⁿ and a^m in full.

aⁿ x a^-n = a^{n+ -n}

= a^n-n
=a¹
=a

Therefore, the negative powers are used to denote reciprocals (the number you multiply n by to get 1 – so 1/6 is the reciprocal of 6 – and it is always 1/n.

So.


3³ =27

3^-3 = 1/27 (or 1/33 – the reciprocal).


Negative powers are easy to manipulate, they just seem a bit weird until you think that positive powers are going up by multiplying the number by itself:

2⁴ is 2 x 2 x 2 x 2

But if you go down, towards 0, the same process is division by two.

64...32...16..8...4..2..

This continues as you go down below zero:

......2, 1, ½, ¼, 1/8, 1/16......

(2¹, 2⁰, 2^-1, 2^-2, 2^-3, 2^-4)

And you can see that these are reciprocals of the powers of 2.

The same applies for the powers of each number.