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?
Best Answer
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 $\;\dfrac{dA}{dd} = 2\pi d.$