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?