3-6-29
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In this problem, we're going to use the product role.
So the derivative of the first times the second plus the
derivative of the second times the first.
So the derivative of three X + 5 to the fifth, we're going to
bring down the five.
We're going to take three X + 5 to the one less power.
Then we have to multiply that by the derivative of that inside
piece, and all that's still going to get multiplied by the
four X + 1 to the negative third.
Now the derivative of the four X + 1 to the negative third, we're
going to bring down the -3 we're going to take it to the one less
power.
Then we have to multiply that by the derivative of the inside,
and that all still is going to get multiplied by three X + 5 to
the 5th.
So the next step we're going to take the derivative of those
inside pieces.
The derivative of three X + 5 is just three.
We have 4X plus one to the -3 still.
So then we're going to have plus that -3 four X + 1 to the -4.
The derivative of four X + 1 is 4, and that's still going to get
multiplied by that three X + 5 to the 5th.
So now we want to take out the exponents that are the smallest.
So we're going to take out three X + 5 to the 4th and we're going
to take out a four X + 1 to the -4.
When we do that, we're going to get left 15 the three times the
five and a 4X plus one from the first term.
When we do the second one, we're going to get -12 times three X +
5.
Now in reality, I could have also taken out of three.
I didn't see that won't matter.
So three X + 5 to the 4th, 4X plus one to the negative 4th.
And then we're going to distribute.
So sixty X + 15 -, 36 X -60.
So when we simplify that up three X + 5 to the 4th, I'm
going to actually move that 4X plus one to the denominator
since it was to a negative exponent.
And then 60 -, 36 X is going to give me 24X and 15 -, 60 is -45.