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 +...

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.