00:01
In this question, we are asked to find the derivative of the function f using the definition of the derivative.
00:06
Recall that by definition, f prime of x equals to the limit as h goes to 0 of f of x plus h, this is an arrow, f of x plus h minus f of x divided by h.
00:26
Where f of x plus h, to get f of x plus h, we simply need to replace x by x plus h.
00:34
In the formula for f.
00:37
We are going to get x plus h plus the square root of x plus h.
00:47
Now let's plug in this expression in the limit.
00:52
So we are going to replace f of x plus h by x plus h plus square root of x plus h.
01:02
And we are going to replace f of x by x plus square root of x.
01:09
And we are going to get that f prime of x equals to the limit.
01:16
As h goes to 0 of x plus h plus square root of x plus h minus x plus square root of x divided by h.
01:35
Now note that we can cancel x here and what we are going to get is the limit as h goes to 0 of h plus square root of x plus h, x cancels minus square root of x divided by h.
02:10
And we can rewrite this as a limit.
02:13
Now, h divided by h equals to 1 plus square root of x plus h minus square root of x divided by h as age as age goes to 0.
02:31
Now we can write this as 1 plus the limit of square root of x plus h minus square root of x over h.
02:48
What we are going to do next is we will rewrite the limit of x plus h.
02:53
As the limit of square root of x plus h minus square root of x over h.
03:04
And now we are going to multiply and divide this expression by square root of x plus h plus square root of x.
03:17
This is the so -called multiplication division by the conjugate...