Section 2.1 Question 1

[hanyu Qi]

Hello everyone,

I am wondering whether I need to show a function is differentible first then calculate the derivative even if the question only ask me to establish the derivative.

E.X.  (sinZ)’ = cosZ

The answer uses the limit definition to show but can I use theorem 3 (C-R equations) to show it as well?

Thank you!

[oighea]

If the question asks you to use the limit definition, use the limit definition.

The sin function is actually derived from the exponential function. We know the natural exponential function is differentiable, and is equal to its own derivative.

$\sin z = \frac{e^{iz} - e^{-iz}}{2i}$
$(\sin z)' = \frac{ie^{iz} + ie^{-iz}}{2i} = i \frac{e^{iz} + e^{-iz}}{2i} = \frac{e^{iz} + e^{-iz}}{2} = \cos z$

Use the limit definition of $e^z$ and the property of limits when regarding addition, multiplication, and composition to obtain the derivative of $\sin$

[wentao huang]

I think some functions are common, whose derivatives are calculated in the previous courses(including high school courses). For example, I think Sinx is an example, x^m is another example. In the practice and test, if question doesn't ask you to use definition, I guess it is not necessary to use that. Meanwhile, the definition method can ensure you to get the correct answer, but it takes some time right?