What is the derivative of # ( cos (pi*x) +1 ) / x#?

1 Answer
Steve M
Nov 4, 2016

# d/dx (cos(pi x)+1)/x= -(pisin(pix))/x - (cos(pix))/x^2 #

Explanation:

If you are studying maths, then you should learn the Quotient Rule for Differentiation, and practice how to use it:

# d/dx(u/v) = (v(du)/dx-u(dv)/dx)/v^2 #, or less formally, # (u/v)' = (v(du)-u(dv))/v^2 #

I was taught to remember the rule in word; " vdu minus udv all over v squared ". To help with the ordering I was taught to remember the acronym, VDU as in Visual Display Unit.

So Let # y=# we have;

# dy/dx = ( (x)(d/dxcos pi x) - (cos pi x)(d/dxx) ) / (x)^2 #
# :. dy/dx = ( (x)(-pisinpix) - (cos pi x)(1) ) / x^2 #
# :. dy/dx = ( -pixsinpix - cos pi x ) / x^2 #
# :. dy/dx = -(pisinpix)/x - (cos pi x)/x^2 #

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