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The distance moved by a particle is described by the $s(t)=\sin^2(3t)$ where $t$ is time measured in seconds. Use successive estimates of the average velocity between $t=0.5$ and $t=0.5+h$, for smaller and smaller $h$ values (e.g: $h=0.1,0.01,0.001,0.0001,\ldots$) to obtain an estimate for the instantaneous velocity of the particle at time $t=0.5$.

I know that first you plug in $t=0.5$ and find that answer, and then you put in $t=0.6, 0.51, 0.501, 0.5001,\ldots$ and you subtract, for example, the answer you get for $t=0.5$ from the answer you get for $t=0.6$ and then you divide them by $0.6 - 0.5$. I have been doing this and I seem to be getting strange answers and would like someone else to try it. Also, I know you can get this answer by finding the derivative and plugging in $t=0.5$.

I am rusty at derivatives, but is the derivative of the function $6\sin(3x)\cos(3x)$?

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