I already know how to take a derivative (the shortcuts such as the power rule, product rule, etc.), but I want to go back to the actual definition of a derivative because I want to apply it to the four types of trig functions. I want to remember how to derive them but can't. The four are a regular trig (the basic six), an inverse trig, the basic hyperbolic, and an inverse hyperbolic trig.

The reason is because I'm not always going to remember the derivatives so I need to be able to derive them long hand. That makes 24 derivatives. The basic 6, the inverse of the basic 6, the six hyperbolic, and the inverse of the six hyperbolic.

Question

I already know how to take a derivative (the shortcuts such as the power rule, product rule, etc.), but I want to go back to the actual definition of a derivative because I want to apply it to the four types of trig functions. I want to remember how to derive them but can't. The four are a regular trig (the basic six), an inverse trig, the basic hyperbolic, and an inverse hyperbolic trig.

The reason is because I'm not always going to remember the derivatives so I need to be able to derive them long hand. That makes 24 derivatives. The basic 6, the inverse of the basic 6, the six hyperbolic, and the inverse of the six hyperbolic.

anonymous · Answer

the problem with that is that you'll have to know (really well) how to use your trig identities and apply them to the limit of the difference quotient (derivative definition).  For the derivative of sine and cosine, all you need to know is lim (as x -> 0) of [sinx/x] = 1 and lim (as x -> 0) of [(cosx-1)/x] = 0.

Just know the derivatives of these two and you'll be able to come up with the derivatives of the other 4 basic trig functions.  Actually this is all I remember myself and I derive the others as needed...