Can Differentiation from First Principles Be Applied to Trigonometric Functions?

[Pseudo Statistic]

Yo.
Curious about two things..
First, is it possible to use a method such as differentiation from first principles (the whole $$lim_{\delta\rightarrow0}\[f(x+\delta)-f(x)]/\delta$$ thing) to differentiate trigonometric functions? Or is everyone stumped with memorizing them? (I'm curious about this, I'm not the kind of guy to memorize a formula list or something, I'm curious as to how these forms were found or an alternative method for finding their derivatives)

Next question, how were the short integration formulae found? Is there anything similar to differentiation from first principles for integration?

Excuse me for my lack of knowledge or ignorance for asking such questions.

Thanks.

 

[Muzza]

http://www.maths.abdn.ac.uk/~igc/tch/ma1002/diff/node27.html

Next question, how were the short integration formulae found?

The what?

 

[Pseudo Statistic]

Formulas, formulae..
Whatever system you go by.

 

[HallsofIvy]

The basic formulae for integration are derived from knowing the corresponding differentiation formulae. For example, since we know
d(xⁿ)/dx= n x^n-1, the integral (anti-derivative) of xⁿ must be (1/(n+1))xⁿ⁺¹ so that ((1/(n+1))xⁿ⁺¹)'= ((n+1)/(n+1))x^(n+1)-1= xⁿ.

I would have thought all that was taught in first semester calculus.

 

[Pseudo Statistic]

I know about that...
Mind you, I haven't taken Calculus yet..
But I guess my second question really had no meaning to it.
Let me add something else, I've noticed:
$$\pi r^2 = \int 2 \pi r dr$$
Is this true for all areas, perimeters and volumes of shapes both 2D and 3D? (Integrating perim. to get area, area to get volume?)

 

[arildno]

I know about that...
Mind you, I haven't taken Calculus yet..
But I guess my second question really had no meaning to it.
Let me add something else, I've noticed:
$$\pi r^2 = \int 2 \pi r dr$$
Is this true for all areas, perimeters and volumes of shapes both 2D and 3D? (Integrating perim. to get area, area to get volume?)

Only sufficiently "nice" regions have this property, for example your circle and also, the sphere.

The cube, with side-length "s", has volume $$s^{3}$$ and surface area $$6s^{2}$$ which doesn't equal the derivative of the volume.

However, in 3-D, we might calculate the volume of a region with the aid of a surface integral using a suitable version of the divergence theorem; similarly, the area of a 2-D figure can be calculated with the aid of a line integral about its periphery with a suitable version of Green's theorem.

 

[Pseudo Statistic]

Would you know of any online resources I could refer to for information on surface integrals?
Thanks.