# Finding f'(x) using definition of derivative

f(x) = x2 cos(1/x)

i know how to take the derivative using product and chain rule, but i need to find the derivative using the definition of the derivative. so far i did:

lim [(x+h)2 cos(1/x + h) - x2 cos(1/x)] / h
h ~> 0

[x2cos(1/x+h) + 2xhcos(1/x+h) + h2cos(1/x+h) - x2 cos(1/x)] / h

then i took [x2cos(1/x+h)- x2 cos(1/x)] and i factored out the x2

x2 [cos(1/x+h) - cos(1/x)]

i used the sum to product formula from trigonometry and i got x2 [-2sin(2x+h / 2x2+2xh)sin(-h / (2x+h / 2x2+2xh)]

but from there i'm stuck. i have no idea how to simplify that expression in order to get the h on the bottom of the the entire fraction to cancel out so i can substitute 0 for h. please help.

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Hao
Instead of using the product formula, it may help to note that we takes the limit of h ->0, so $$h / x$$ is small. This suggests simplification of the argument $$\frac{1}{(x+h)} = (x+h)^{-1}$$ through binomial expansion.