Derivative of `tan^(3)theta` with respect to `sec^(3)theta` at `theta=(pi)/(3)` is

Options

A. `(3)/(2)`
B. `(sqrt(3))/(2)`
C. `(1)/(2)`
D. `-(sqrt(3))/(2)`

Explanation

Given `y = tan ^(3) theta, x sec^(3) theta`
Differentiate both x and y w.r.t. `theta`
`(dy)/(d theta)= 3 tan^(2) theta sec^(2) theta`.
`(dx)/(d theta)= 3 sec^(2) theta xx sec theta tan theta`
`(dy)/(dx)=(dy)/(d theta)xx(d theta)/(d)`
`=(3 tan^(2) theta xx sec^(2) theta)/( 3 sec^(2) theta xx sec theta tan theta)`
`=(tan theta)/(sec theta)=( sin theta)/(cos theta)xxcos theta `
`sin theta`
At `theta =(pi)/(3)`
`(dy)/(dx)= sin ((pi)/(3))= (sqrt(3))/(2)`
Hence, correct answer from the given alternatives is (b).