click to play button
click to replay button
3-2-13
X
    00:00 / 00:00
    CC
    Looking for the derivative. So the limit as H approaches zero of the quantity three of the quantity X + H + 5 / X + H minus the quantity three X + 5 / X. So the limit as H goes to zero. We're just going to do some algebra here. Distribute the three. So 3X plus three H + 5 / X + H, distributing the negative -3 X -5 / X. So the 3X and the -3 X cancel. So we get the limit as H goes to zero of three H + 5 / X + H -, 5 X all over H Then we're going to separate the three H / H portion out because three H / H is just three. The limit as H goes to 03 is 3, so we really need to look at what happens to that second part. So the limit is H goes to 0 of 5 / X + H -, 5 / X all over H So we're going to get a common denominator, and that's going to be X + H * X. So we get five X -, 5 times the quantity X + H / X + H * X all over H So then distributing the -5, the 5X and the -5 X are going to cancel and we get the limit as H approaches zero of the -5 H right here of X + H divided by XH. The H is all now cancel. So we have that three from before. Plus the limit is H goes to 0 of -5 / X + H * X. When H goes to zero, we end up with 3 - 5 / X ^2. The second part says what happens or what's the slope at the tangent line X equal to. So at X equal to, we literally are just going to stick 2IN for the X back on the derivative 3 -, 5 / 2 ^2 3 -, 5 fourths, 12 fourths -5 fourths, which would give us 7 fourths.