U1S4V12 Strategy.txt

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We gave you this function f and we
said that we wanted to calculate f prime of 1.
And we asked you, should we plug in x equals 1 first
and then differentiate?
Or should we take the derivative first and then plug in?
Does it even matter?
Well, there's a right way and there's a wrong way.
The wrong way is plugging in x equals 1 first.
If we did that, well, then we'd get f of 1 equals minus 4.
And that's fine.
That's true.
But that's not really that much information.
So if we were to graph that, then that
tells us that the point 1, minus 4 is on the graph of f.
But knowing one point is not going to tell us
a slope of a tangent line.
So that's the wrong way.
The right way is to differentiate first.
So let's just calculate f prime of x in general.
f is a sum, and we know that the derivative of a sum
is the sum of derivatives.
So f prime of x is just going to be
equal to the derivative of minus 3x squared.
So I'm just going to write that as minus 3x squared prime.
Plus the derivative of 1 over x, so 1 over x prime.
Plus minus 2 prime.
And then we just need to evaluate each of these.
So for minus 3x squared, well, we
have this coefficient out in front.
And we know that when you have a constant times a function
and you want to differentiate it,
you can just keep the constant out in front.
So we'll have the minus 3 times x squared prime.
And then we're adding that to the derivative of 1 over x.
That we've already calculated.
That's going to be minus 1 over x squared.
And then for the derivative of negative 2, negative 2
is a constant.
And the derivative of a constant just the by itself is 0.
So putting all of this together, we're
getting minus 3 times and then the derivative of x
squared we've already done.
That's 2x.
Minus 1 over x squared.
And that's minus 6x minus 1 over x squared.
So that's f prime of x.
Now we get to plug in.
If we want f prime of 1, then we're
just going to get minus 6 minus 1, which is minus 7.
And that's our answer.