How do I use the power rule to find the derivative?

Question

y = πsinx - 4cosx

Answer 1

by power rule, i assume you mean an exponential (taylor/maclaurin) series..

if that's the case:

y = pi * sin x - 4 cos x

we separate this into the fourth-order exponential approximations of these functions

= pi*(x - x^3 / 3! + x^5 / 5! - x^7 / 7! ...) - 4*(1 - x^2 / 2! + x^4 / 4! - x^6 / 6! ...)

we now take the derivatives of the separate functions (using the rule d/dx x^n = nx^(n-1) )

y' = pi*(1 - 3x^2 / 3! + 5x^4 / 5! - 7x^6 / 7! ...) - 4*(0 - 2x / 2! + 4x^3 / 4! - 6x^5 / 6! ...)

now since the coefficient of x^n in the numerator of each fraction is taken as a factorial on the denominator, we can reduce the factorial by 1 and cancel the number from the numerator

= pi*(1 - x^2 / 2! + x^4 / 4! - x^6 / 6! ...) - 4*(- x + x^3 / 3! - x^5 / 5! ....)

which we recognise as:

y' = pi* cos x - (-4 *sin x)

= pi*cos x + 4*sin x

so there's your answer. i apologise for the slightly messy notation, but it's as good as i can do with y!a

Answer 2

This isn't a "power rule" problem, since these functions have a different derivative than do terms like a*x^n . Here, you use the fact that
d(sinu)/du=cos u and d(cosu)/du= -sin(u).
So the derivitative is pi*cos(x) + 4 sin(x)