We can differentiate term by term. (Things that are added together are called "terms".)
So we need to differentiate 1/2t^612t6 and -3t^4−3t4 and 11. (We don't need to write all of this, but I'm explaining our thought process.)
To differentiate 1/2 t^612t6 we will used the power rule. The constant 1/212 just hangs out out front.
In more proper language: the derivative of 1/212 times t^6t6 is 1/212 times the derivative of t^6t6.
In notation: d/dt(1/2t^6) = 1/2 d/dt(t^6)ddt(12t6)=12ddt(t6)
Now, we find the derivative of t^^6t∧6. Multiply by the exponent, then subtract one from the exponent to get the new exponent.
6t^(6-1) = 6t^56t6−1=6t5
We use the same process to differentiate -3t^4−3t4.
The derivative of a constant, like 11, is 00
The derivative of f(t) = 1/2t^6-3t^4+1f(t)=12t6−3t4+1 is
f'(t) = 1/2(6t^5) - 3(4t^3) +0 " " (we often skip writing this)
f'(t) = 3t^5-12t^3
