find the rate of change of circumference of a circle with respect to it's radius.
(a) π
(b) 2π
(c) 2πr
(d) 3π
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find the rate of change of circumference of a circle with respect to it's radius.
(a) π
(b) 2π
(c) 2πr
(d) 3π
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Question :-
Find the rate of change of circumference of a circle with respect to it's radius.
(a) π
(b) 2π
(c) 2πr
(d) 3π
[tex]\large\underline{\sf{Solution-}}[/tex]
Let assume that radius of circle be 'r' units.
Let assume that C represents the circumference of a circle.
We know,
Circumference, C of a circle of radius r is given by
[tex]\rm \: C \: = \: 2 \: \pi \: r \\ [/tex]
On differentiating both sides w. r. t. r, we get
[tex]\rm \: \dfrac{d}{dr}C \: = \dfrac{d}{dr}[\: 2 \: \pi \: r ]\\ [/tex]
[tex]\rm \: \dfrac{d}{dr}C \: = 2 \: \pi \: \dfrac{d}{dr}r \: \\ [/tex]
[tex]\rm \: \dfrac{d}{dr}C \: = 2 \: \pi \: \times 1 \: \\ [/tex]
[tex]\rm \: \dfrac{dC}{dr} \: = 2 \: \pi \: \: \\ [/tex]
Hence,
[tex]\rm\implies \:\boxed{\sf{ \: \: \rm \: \dfrac{dC}{dr} \: = 2 \: \pi \: \: }}\\ [/tex]
So, option (b) is correct.
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Additional Information :-
[tex]\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \dfrac{d}{dx}f(x) \\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf 0 \\ \\ \sf sinx & \sf cosx \\ \\ \sf cosx & \sf - \: sinx \\ \\ \sf tanx & \sf {sec}^{2}x \\ \\ \sf cotx & \sf - {cosec}^{2}x \\ \\ \sf secx & \sf secx \: tanx\\ \\ \sf cosecx & \sf - \: cosecx \: cotx\\ \\ \sf \sqrt{x} & \sf \dfrac{1}{2 \sqrt{x} } \\ \\ \sf logx & \sf \dfrac{1}{x}\\ \\ \sf {e}^{x} & \sf {e}^{x} \end{array}} \\ \end{gathered}[/tex]