00:01
Okay, so in this question, first of all, we're being asked to find the derivative of this function using the definition.
00:10
So we've got f of x equals minus 2x square plus 1, and we're asked to find its derivative at the point 2 minus 7.
00:31
So first of all, let's remember that the definition of the derivative is that f -dash -x is the limit as t goes to zero of f of x of f -of -t -minus f of x divided by t.
00:57
So let's apply this definition.
01:01
So we want to take the limit as t goes to zero of minus 2 x plus t squared plus 1 plus 2x squared minus 1 divided by t.
01:17
So these ones are going to cancel out straight away.
01:21
And let's keep going with the algebra.
01:26
1 over t.
01:28
So minus 2 x plus t squared.
01:32
So we're going to get minus 2x squared.
01:34
And then we're going to get 2xt times minus 2x2 gives us minus 4xt.
01:40
And then minus 2t squared plus 2x squared.
01:48
So these cancel.
01:50
And then we're taking the limit as t goes to zero, which means that let's do this 1 over t first.
01:59
So this gives us the limit.
02:01
As t goes to 0 of minus 4x because i'm doing 1 over t here minus 2t but when t goes to 0 this goes to 0 so we get minus 4x so that's f dash of x and then we're applying this at the point where x is 2 so the derivative at x equals 2 is minus 8 so that's the answer to that question now question 2.
02:34
We're given this horrible function.
02:38
F of x equals x cubed plus the cube root of 5x plus 3 minus root x plus 3 and then i think it's add 2.
03:01
And we want to find its derivative at the point p which is 1 3.
03:10
So x equals 1, f of x equals 3.
03:14
So let's do the derivative.
03:17
F dash of x is 3x square, because remember we take this number here, we put it in front, and then we would decrease this number by 1.
03:27
So this we can write as 5x plus 3 to the power of 1 third.
03:34
So now i'm going to apply that same logic to this.
03:37
So we get plus a third, 5x plus 3 to the minus 2 thirds.
03:46
And then here, this is x plus 3 to the 1 half.
03:51
So minus a half, x plus 3 to the minus a half.
03:56
And then the derivative of 2 is 0.
04:00
So now we're going to put in that this is at the point where x equals 1.
04:04
So we get 3 plus 1 third times.
04:09
8 to the minus 2 thirds, minus a half times 4 to the minus a half, which tells us that the derivative here is 17 over 6, which in decimals is about 2 .8 .3.
04:35
Okay.
04:37
Now the next part of this question, 3f of x is equal to 3x square plus 4x plus 3.
04:54
So first of all, we find the derivative.
04:57
So we take this number out the front, 6x.
05:02
Here we just get 4...