2. Derivatives of Csc, Sec and Cot Functions
by M. Bourne
By using the quotient rule and trigonometric identities, we can obtain the following derivatives:
`(d(csc x))/(dx)=-csc x cot x`
`(d(sec x))/(dx)=sec x tan x`
`(d(cot x))/(dx)=-csc^2 x`
In words, we would say:
The derivative of `csc x` is `-csc x cot x`,
The derivative of `sec x` is `sec x tan x` and
The derivative of `cot x` is `-csc^2 x`.
If u = f(x) is a function of x, then by using the chain rule, we have:
`(d(csc u))/(dx)=-csc u\ cot u(du)/(dx)`
`(d(sec u))/(dx)=sec u\ tan u(du)/(dx)`
`(d(cot u))/(dx)=-csc^2u(du)/(dx)`
Example 1
Find the derivative of s = sec(3t + 2).
Example 2
Find the derivative of `x = θ^3 csc 2θ`.
Example 3
Find the derivative of y = sec43x.
Exercises
1. Find the derivative of y = csc2(2x2).
2. Find the derivative of y = sec2 2x.
3. Find the derivative of 3 cot(x + y) = cos y2.
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