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.

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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:



The denominator tends to `x^2` , the numerator is equal to



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.


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