Why Is The Derivative Of Sinx = Cosx?

The derivative of any function f(x) can be found from first principles as the difference between two values of f(x) infinitely close to each other divided by the infinitesimal difference in x between the two f(x) values. I.e.

f`(x) = lim (h->0) ((f(x+h) - f(x)) / h) . The trick is to know when to sub in 0 for h, as doing this too early will lead to division by 0, a maths error.

d/dx (sinx) = lim(h->0) ((sin(x+h) - sinx) / h) . Manipulating the numerator:

sin(x+h) - sinx = sinxcos(h) + sin(h)cosx - sinx, using the sin(A+B) addition formula.

For small x, sinx ≈ x, cosx ≈ 1-(x^2)/2. h is infinitesimal hence:

sinxcos(h) + sin(h)cosx - sinx = sinx(1-(h^2)/2) + hcosx - sinx =

= sinx - ((h^2)/2)sinx + hcosx - sinx =

= - ((h^2)/2)sinx + hcosx. Now h can be cancelled out from the derivative`s numerator and denominator. Note: this is allowed because h is not 0, but rather infinitesimal.

d/dx (sinx) = lim(h->0) (cosx - (h/2)sinx)

= cosx - (0/2)sinx

d/dx (sinx) = cosx, as required.

Exercise for the reader: what is the derivative of cosx? Hint: sketch the graphs of the trigonometric functions and pay attention to the positions of plateau and intervals of maximum gradient on the cosine graph. Once you think you might know what it is, try to prove it through first principles.