00:01
In this problem, we will cover the tangent line of a function.
00:04
So we want to find the tangent line equation for this function at the point 1 -25th, and to do that, we must first find the slope of the tangent line.
00:16
And to find the slope of the tangent line, we must use the derivative.
00:20
So i will start off by writing f prime of 1 is equal to the limit as x approaches 1 of f of x, minus f of 1 over x minus 1.
00:36
Now we just substitute for f of x and f of 1, and we get 2 over x plus 4 minus 2 5th, and that's over x minus 1.
00:53
And looking strictly at the top, we want to combine these two fractions, as we may call them.
01:01
So i'm going to work at the side in green.
01:05
And if we were to multiply the first fraction by 5 over 5, and the second fraction by x plus 4 over x plus 4, we will get, let's see, 10 minus 2x minus 8, and that's over 5 times x plus 4, which is going to be 5x plus 20.
01:32
And this can be simplified to negative 2 times x minus 1 over 5x plus 20.
01:46
So now that's the simplified version of the top, and we still have x minus 1 at the bottom.
01:52
So we're going to do negative 2 times x minus 1 over 5x plus 20 times 1 times 1 over x plus 20 times 1 over x minus 1...