$${\frac{d}{dr}}(\pi r^2)=2\pi r$$
Why is that? And additionaly: Why doesn't hold this for the volume and area of a circle?
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Could you be more specific when you say "Why doesn't hold this for the volume and area of a circle?" – Simply Beautiful Art Oct 01 '17 at 16:46
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2It does hold: $\dfrac{\mathrm d}{\mathrm dr} \dfrac43 \pi r^3 = 4 \pi r^2$ – Kenny Lau Oct 01 '17 at 16:46
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As for nice and pretty pictures as to why this holds, see 3blue1brown's calculus videos This particular time stamp deals with why$$\int_0^R2\pi r~\mathrm dr=\pi R^2$$but you can apply the reasoning backwards. – Simply Beautiful Art Oct 01 '17 at 16:47