Understand the power rule in differentiation.

Decreasing power with derivatives

As fun as it is to work out derivatives using deltas like Δx\Delta x, and seeing what happens when we make them smaller and smaller, we can often do it without doing all that work.

Let’s see if we can decipher a pattern in the derivatives we’ve worked out so far:

s=t2st=2ts = t^2 \rightarrow \frac {\partial s}{\partial t} = 2t

s=t2+2tst=2t+2s = t^2 + 2t \rightarrow \frac {\partial s}{\partial t} = 2t + 2

s=t3st=3t2s = t^3 \rightarrow \frac {\partial s}{\partial t} = 3t^2

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