By making use of the quotient rule and also trigonometric identities, we can achieve the adhering to 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 certainly say:

The derivative of `csc x` is `-csc x cot x`, The derivative of `sec x` is `sec x tan x` and also The derivative the `cot x` is `-csc^2 x`.

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If u = f(x) is a role of x, then by utilizing the chain rule, us 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).

Answer


Put `u = 3t + 2`. Then:

`s=sec u` and

`(du)/(dt) = 3`

So using Chain Rule, us have:


`(ds)/(dt) =(ds)/(du) (du)/(dt)`

`=sec(u) tan (u)(3)`

`=3 sec(3t+2) tan (3t+2)`


Example 2

Find the derivative that `x = θ^3 csc 2θ`.

Answer


` x=theta^3csc 2 theta `

If us let `u=theta^3` and also `v=csc 2 theta`, climate


`(dx)/(d theta) =u(dv)/(d theta)+v (du)/(d theta)`

`=theta^3(-csc 2 theta cot 2 theta)(2)+` `csc 2 theta(3 theta^2)`

`=theta^2(csc 2 theta)(-2 theta cot 2 theta+3)`


Example 3

Find the derivative of y = sec43x.

Answer


`y=sec^4 3x`

Let `y=u^4`, wherein `u=sec 3x`.

Then


`(dy)/(dx)=(dy)/(du)(du)/(dx)`

`=4u^3(sec 3x tan 3x)(3)`

`=4(sec^3 3x)(sec 3x tan 3x)(3)`

`=12 sec^4 3x tan 3x`


Exercises

1. Discover the derivative the y = csc2(2x2).

Answer


This is an example of a function of a role of a function, and also we require to apply chain dominance 3 times.

Let u = 2x2 and also v = csc u.

So y = v2


` (dy)/(dx)=(dy)/(dv)(dv)/(du)(du)/(dx)`

`=<2v><-csc u cot u>(4x)`

`=<2 csc(2x^2)>` `xx<(-csc 2x^2)(cot 2x^2)>(4x)`

`=-8x(csc^2 2x^2)(cot 2x^2)`


2. Find the derivative of y = sec2 2x.

Answer


This is additionally an example of a duty of a role of a function, and also we require to use chain dominion 3 times.

This have the right to be composed `y = sec^2u` wherein `u = 2x`.

If us let `v = sec u` climate `y=v^2`.

So us have:

`y=v^2=sec^2 u`

Then


`(dy)/(dx)=(dy)/(dv)(dv)/(du)(du)/(dx)`

`=(2v)(sec u tan u)(2)`

`=(4v)(sec u tan u)`

`=(4 sec u)(sec u tan u)`

`=4 sec^2 u tan u`

`=4 sec^2 2x tan 2x`


3. Discover the derivative of 3 cot(x + y) = cos y2.

Answer


This is an latent function.

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3 cot(x + y) = cos y2

For the left hand side, we put u = x + y.

Differentiating 3 cot u gives us:

`3(-csc^2 u)((du)/(dx))`

Substituting because that `u` and performing the `(du)/(dx)` part gives us:

`-3 csc^2(x+y)(1+(dy)/(dx))`

On the ideal hand side, us let u = y2. Separating `cos u` gives us:

`(-sin u)((du)/(dx))`

Substituting because that `u` and performing the `(du)/(dx)` part offers us:

`(-sin y^2)(2y(dy)/(dx))`

We placed both political parties together:

`-3 csc^2(x+y)(1+(dy)/(dx))` `=(-sin y^2)(2y(dy)/(dx))`

Expanding gives:

`-3 csc^2(x+y)` `-3 csc^2(x+y)(dy)/(dx)` `=-2y sin y^2(dy)/(dx)`

Adding

`2y sin y^2(dy)/(dx)`

to both sides:

`-3 csc^2(x+y)` `-3 csc^2(x+y)(dy)/(dx)` `+2y sin y^2(dy)/(dx)` `=0`

Adding

`3 csc^2(x+y)`

to both sides:

`-3 csc^2(x+y)(dy)/(dx)` `+2y sin y^2(dy)/(dx)` `=3 csc^2(x+y)`

Factoring out the dy/dx term:

`<2y sin y^2-3 csc^2(x+y)>(dy)/(dx)` `=3 csc^2(x+y)`

This provides us:

`(dy)/(dx)` `=(3 csc^2(x+y))/(2y sin y^2` `- 3 csc^2(x+y)`


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