How do you differentiate #f(x)= ( x +2sinx )/ (x + 4 )# using the quotient rule?

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Answer 1 · 2016-06-03T01:25:59

#f'(x)=(2(cosx(x+4)-sinx+2))/(x+4)^2#

The quotient rule states that the derivative of a function expressible as the quotient of two other functions

#f=(g/h)#

Has a derivative of

#f^'=(g^'h-gh^')/h^2#

For #f(x)=(x+2sinx)/(x+4)#:

We see that #g=x+2sinx# so #g^'=1+2cosx# and #h=x+4# so #h^'=1#.

This gives:

#f'(x)=((1+2cosx)(x+4)-(x+2sinx)(1))/(x+4)^2#

#f'(x)=(x+4+2xcosx+8cosx-x-2sinx)/(x+4)^2#

#f'(x)=(2(cosx(x+4)-sinx+2))/(x+4)^2#