Essential Knowledge 2.1A4

Essential Knowledge 2.1A4 Students will know that for y = f(x), notations for the derivative include , f '(x), and y'.

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In the previous Essential Knowledge, we were given the equation of an object dropped from 1000 feet above ground.

y = -16t² + 1000

We wanted to find the velocity the ball would be falling at, which we realized would be the instantaneous rate of change, which we learned is called a "derivative". We found the derivative of this equation using the limit of a difference quotient, and we got.

v = -32t

Now what if we weren't talking about positions and velocities, and we were just told "find the derivative of the equation"

y = -16x² + 1000

Well, we would get -32x for our derivative, but what would it be equal to?

___ = -32x

We could write

The derivative = -32x

But of course in math we like to use symbols. There are a few different symbols we use. My favorite is:

You would pronounce this "d y d x equals negative thirty-two x"

The official meaning is "The derivative of y with respect to x" = -32x

When you see this you can think: "The derivative of an equation where y was the isolated dependent variable and x was the independent variable is -32x." I like this symbol because it makes it easier to perform integrals (our third and final big idea in AP Calculus AB)

Another way you can write derivative is with "primes"

y' = -32x

You would pronounce this "y prime equals negative thirty-two x"

The official meaning is "The derivative of y" = -32x

When you see this you can think: "The derivative of an equation where y was the isolated dependent variable is -32x." You'll notice the drawback of this notation is that you lose information about which variable the derivative used. Usually you would use prime notation for easier derivatives. This derivative is appropriate, because it's kinda obvious when you see the x chilling with the -32 that you probably took the derivative of y with respect to x.

If the equation was instead written as a function

f(t) = -16t² + 1000

You would use prime notation, but written a little differently

f'(t) = -32t

All notations simply let you know that a derivative has been taken.