Differentiate f(x)= (1/x)sinx by using the definition of the derivative. Hints: You may use any of the following: sin(x+h)=sinxcos h + cosxsinh and lim_(x-gt0) sinx/x = 1 and lim_(x-gt0) (cosx-1)/x = 0. I need help with this question.

Hello!
By the definition of the derivative we need to find the limit of (f(x+h)-f(x))/h for h-gt0 and any fixed x. Consider the difference:
f(x+h)-f(x)=(sin(x+h))/(x+h)-(sin(x))/x=
=1/(x(x+h))*(x*sin(x+h)-(x+h)*sin(x)).
The denominator tends to x^2 , the numerator is equal to
x*sin(x)cos(h)+x*cos(x)sin(h)-x*sin(x)-h*sin(x)=
=x*sin(x)(cos(h)-1)+x*cos(x)sin(h)-h*sin(x).
Dividing this by h as required and using the given limits we obtain for h-gt0
x*sin(x)(cos(h)-1)/h+x*cos(x)sin(h)/h-sin(x) -gt 0+x*cos(x)-sin(x).
Recall the denominator x^2 and the derivative is
(x*cos(x)-sin(x))/x^2,  which is correct.