Essential Knowledge 2.1C1

Essential Knowledge 2.1C1 Students will know that direct application of the definition of the derivative can be used to find the derivative for selected functions, including polynomial, power, sine, cosine, exponential, and logarithmic functions.

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We're almost to the shortcuts. This objective requires you to solve without shortcuts though. This is probably good, since all of the shortcuts were found this way. For now, you're still just going to have to use the difference quotients.

or

From what I've been seeing, the x&h one seems to give easier answers. Anyways, mathematicians have found shortcuts for all of the different types of functions we normally use

Polynomial

These are the ones I've done so far. See EK2.1A2 or EK2.1B1. Basically you just have to find a way to factor in order to cancel out the denominator. And there is always a way.

Power

A polynomial function is a bunch of power functions added together, so this is like we've been doing except way easier.

Sine

Here is how you find the derivative of sine. Turns out it's cosine. What a nice shortcut that will be!

Cosine

Here is how you find the derivative of cosine. Turns out it's negative sine. What a nice shortcut that will be!

Exponential

Usually when someone says exponential they mean something with e having a variable exponent. Of course an example would be e^x. The derivative of e^x is e^x. Yep you read that right. In fact, the number e was invented for this to be true.

Unfortunately exponential could also mean a number other than e, usually written as a^x. I'm sure there must be a way to solve this without the shortcut, but I don't know it. And really no one cares. You almost never use this, I'd be surprised if it came up on the AP exam. Either way, we'll learn the shortcut and that's the only thing you should care about.

Logarithmic

No need to know how to do this without the shortcut.