- #1
demonelite123
- 219
- 0
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.
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.