Hello!
Given a function `f(x)=x sin(x),` and we have to find its derivative by the definition. Consider the expression `(f(x+Delta x)-f(x))/(Delta x)` and find its limit for `Delta x->0:`
`(f(x+Delta x)-f(x))/(Delta x)=((x+Delta x)sin(x+Delta x)-x sin(x))/(Delta x)=`
`=(x(sin(x+Delta x)-sin(x))+Delta x sin(x+Delta x))/(Delta x)=`
`=x(sin(x+Delta x)-sin(x))/(Delta x)+ sin(x+Delta x).`
The second summand has the limit `sin(x),` because `sin(x)` is continuous for any `x.`
The first summand has the limit `x(sin(x))'=xcos(x)` (by the definition of the derivative, applied to `sin(x)`).
So the answer is `(x sin(x))'=x cos(x)+sin(x).`
If we "don't know" what the derivative of `sin(x)` is, transform the expression:
`(sin(x+Delta x)-sin(x))/(Delta x)=(2sin((Delta x)/2) cos(x+(Delta x)/2))/(Delta x)=`
`=sin((Delta x)/2)/((Delta x)/2) * cos(x+(Delta x)/2).`
The second factor has the limit `cos(x)` because `cos(x)` is continuous for any `x,` the first has the limit 1 (we should know this).
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