In this tutorial, we're going to look at differentiating x to the
power n from first principles.
Now n could be a positive integer, n could be a fraction.
It could be negative, or it could be 0.
So we're going to start by taking the case where n is a positive integer.
So we'll be looking at things like x squared, x to the power 7.
Even x to the power 1.
So we have y equals x to the power n.
Our definition of our derivative function is dy by dx equals
the limit as delta x approaches 0, of f of x + delta x,
minus our function of x, divided by delta x.
So let's just look at this part first of all. Our f of x + delta x
is going to equal x + delta x, all to the power n.
And this is a binomial.
So what we're going to start by doing is actually just expanding
a + b to the power n.
And that is a to the power n,
plus n, times a to the power of n - 1, multiplied by b,
plus and there are lots of other terms in between both containing powers of a and b
along to our final term, which is b to the power n.
Well, now let's look at what we've got.
Instead of our a, we've got x, and instead of the b, we've got delta x.
So, we've got x + delta x to the power n.
So our a, we've got x to the power n,
plus n x to the power n - 1 and our b is delta x plus, and
again, all these terms which will be in terms of x and delta x
Up to our last term, which is delta x to the power n.
Now let's substitute this in into our derivative here.
So our dy by dx. is the limit as delta x approaches 0,
of our function of x + delta x.
Which is x to the power n, plus, n x to the n - 1, delta x, plus,
and again all these terms in both x and delta x, plus delta x to the power n.
Minus our function of x, which is x to the power n.
All divided by delta x.
Oh, here we have x to the power n takeaway x to the power n.
So our derivative is the limit as delta x approaches 0.
Of n x to the n - 1, delta x plus all the terms in x and delta x.
Plus delta x to the power n. All divided by delta x.
Now all these terms, have delta x in them.
And we're dividing by delta x, so we can actually cancel the delta x.
So that we have the limit as delta x approaches 0.
Of n x to the power of n -1, plus, now all these terms here
have delta x squared, delta x cubed and higher powers all the way up
to delta x to the power n. So when we've divided by delta x,
all of these have still got a delta x in them up to our
final term of delta x to the power of n -1.
So when we actually take the limit of delta x approaches 0.
All of these terms are going to approach 0.
They've all got delta x in them so they'll all be 0.
So our derivative is just n times x to the power of n - 1.
So at derivative of x to the power n,
is n times x to the power of n - 1.
Let's have a look at some examples.
Let's say y equals x squared.
So our dy by dx equals
Well, the power comes down in front, so it's
2 times x to the power of 2 take away 1 which gives us
2 times x to the power 1, which is just 2x.
y equals x to the power 7.
dy by dx equals... the power comes down
7 times x to the power of 7 - 1, so we get 7x to the power of 6.
Now.
What happens when we have y equals just x?
x to the power of 1?
Well, we already know the derivative of this, but let's
see how it fits with the rule.
Bring down the power 1 times x to the power of 1 take away 1,
so we have 1 times x to the power of 0.
Well, x to the power of 0 is 1, so it's 1 times 1, so
we end up with the derivative of 1, which is
exactly what we expected because we know the
gradient function of y equals x is 1.
So that was when x was a positive integer. What happens when x is 0?
Well, let's have a look: y equals x to the power 0.
Well, we just saw here that x to the power 0 is actually 1.
And if we find the derivative of y equals 1. Well, y equals 1 is a
horizontal line, so the derivative is 0.
So y equals x to the power 0 when n is 0. The derivative is 0.
Let's have a look at some more complicated examples.
Let's try y equals 6 x cubed, minus 12 x to the power 4, plus 5.
dy by dx is equal to...
If that's 6 multiplied by 3 as we bring the power down,
x to the power, take one from the power,
minus 12 times, bring the power down, 4,
x to power of 4 take away 1 and our derivative of 5 is 0.
(I'll put the plus 0 there.)
So three 6s... 18 x to the power 3 - 1 is 2
minus four 12s... 48 x to the power 4 - 1 is 3.
So there's our derivative.
Let's try another one.
y equals x minus 5x to the power 5 + 6 x to the power 7 + 25.
So our derivative dy by dx equals... Now this is x to the power 1.
So, 1 times x to the power 1 - 1,
minus 5 times, bring the power down, 5 times x to the power of 5 - 1,
plus 6 times, bring the power down, 7 multiplied by x to power 7 - 1,
plus the derivative of 25, again 0.
So we have 1 times x to the power of 0, which is just 1,
minus, five 5s are 25, times x to the power 5 - 1 is 4,
plus six 7s, 42 times x to the power 7 - 1 which is 6.
Now we've proved this result when n is a positive integer,
but it actually works also when n is a fraction or when it's a
negative number. Now we're not going to do the proof of this
because it requires a more complicated version of the
binomial expansion, but we're still going to use the result,
so let's have a look at y equals x to the power a half.
Our dy by dx is going to be a half, as we bring down the power,
x to the power of a half take away 1 which equals
a half times x, and a half take away 1, is minus a half.
Let's have a look now, when n is a negative number, so
y equals x to the power of minus 1. So dy by dx equals, let's bring the power down,
minus 1 times x to the power of minus 1 take away 1, so we have...
Minus x to the power -1 - 1 is -2.
Now, before we finish, let's look at two more complicated
examples where we need to do a little bit of rewriting in
index notation before we can carry out the differentiation.
So let's have a look at y equals 1 over x plus 6x
minus 4x to the power of 3 over 2 plus 8.
Now we need to rewrite this in index notation so that we can
easily differentiate it. So that's x to the power of minus 1
plus 6x minus 4x to the power 3 over 2 plus 8.
So let's differentiate dy by dx equals...
Bring the power down, minus 1 x to the power of minus 1 take away 1,
plus derivative of 6x to the power 1, that's 6 times x to the power 1 - 1,
minus... now 4 times 3 over 2,
multiplied by x to the power of 3 over 2 take away 1,
plus, the derivative of 8, which is 0.
So here we have minus x, -1 - 1 is -2.
Plus 6 x to the power 1 - 1 is 0 which is 1, so it's just plus 6.
Minus... three 4s are 12 divided by 2 gives us 6,
x to the power of 3/2 minus 1...
So that's 1 and a half minus 1, so we end up with x to the power a half.
And there's our derivative.
OK, one more example, y equals 4x to the power of one third,
minus 5x plus 6 divided by x cubed.
Now again, we've got a mixture of notations and to
differentiate it we need to write it all in index notation
rather than having the division. So this will be 6x to the power of -3,
So, our dy by dx equals...
4 multiplied by the power, which is a third,
x to the power of 1/3 take away 1,
minus, this is 5 x to the power 1, so it's 5 times 1,
multiplied by x to the power of 1 - 1,
plus 6 multiplied by minus 3 times x to the power of -3 - 1.
So let's tidy it all up.
We get 4 thirds x to the power 1/3 take away 1 is minus 2/3,
so it's x to the power of minus 2/3,
minus, now here x the power of 1 - 1 is x to the power 0, which is 1,
so it's just minus 5,
Six times -3 is -18 and x to the power -3 - 1 is -4.
So, there's our derivative.