Looks good!
Hey guys, I'm using the definition of derivative to find f'(a), this always makes a mess. This is what I have done so far;
Question: f(t) = 2t^{3}+t, find f'(a).
So,
lim f(a+h)-f(a)/(h) - the formula
h->0
lim 2(a+h)+(a+h)-(2(a)^{3}+a)/(h)
h->0
Here's all the foiling etc I did on the side:
(a+h)(a+h)(a+h)
(a^{2}+2ah+h^{2})(a+h)
eventually getting....2(a^{3}+3a^{2}h+3ah^{2}+h^{3}) = 2a^{3}+6a^{2}h+6ah^{2}+2h^{3}
so now I have,
lim 2a^{3}+6a^{2}h+6ah^{2}+2h^{3}+a-h-2a^{3}-a/(h)
h->0
lim 6a^{2}h+6ah^{2}+2h^{3}-h/(h)
h->0
lim 6a^{2}h+6ah^{2}+2h^{3}
h->0
6a^{2}(0)+6a(0)^{2}+2(0)^{3}-1 = -1
f'(a) = -1?
Not sure where I went wrong, the derivative of 2t^{3}+t is definitely not -1. Could use some guidance.
Thank you.
Ok so I figured out that I missed a step, after
lim 6a^{2}h+6ah^{2}+2h^{3}-h/(h)
h->0
I should have isolated h so I get
lim h(6a^{2}+6ah+2h^{2}-1)/h (now cancel out the h's)
h->0
lim 6a^{2}-6ah-2h^{2}+1
h->0
6a^{2}-0-0+1
= 6a^{2}+1 how does this look?