How would I solve the following problem?

Find the rate of change of surface area of a sphere with respect to its diameter D.

I know the formula for surface area of a sphere is

$A=4\pi r^2$

So I know the rate of change of area with respect to radious is $\frac{dA}{dR}$

but how would I find find it with respect to diameter?

  • $\begingroup$ Do you know the definition of the diameter ? $\endgroup$ – Damien L Feb 14 '13 at 19:52
  • $\begingroup$ yes it is two times the radius. $\endgroup$ – Fernando Martinez Feb 14 '13 at 19:55

Hint: $r = \dfrac 12 d.\quad$ (The diameter of a sphere is twice the length of the radius.)

That gives you $$A = 4\pi\left(\frac 12 d\right)^2 = 4\pi \left(\frac 14\right) d^2 = \pi d^2$$

Now you can find $\;\dfrac{dA}{dd} = 2\pi d.$

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  • $\begingroup$ Yes you are right, I got $2dpi$ as my derivative. $\endgroup$ – Fernando Martinez Feb 14 '13 at 19:59
  • $\begingroup$ Great, Fernando...indeed! You've got it. $\endgroup$ – amWhy Feb 14 '13 at 20:05

$A=4\pi r^2=\pi D^2$, so $\frac{dA}{dD}=2\pi D$.

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