# What is the derivative of #1/(sec x - tan x)#?

Note that:

Simplifying:

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Mason has given a fine answer, here's a bit of a tricky one.

One of the Pythagorean trigonometric identities is

Now use the differentiation rulles for these trig functions to get

Comparing answers:

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To find the derivative of ( \frac{1}{\sec(x) - \tan(x)} ), apply the quotient rule.

Let ( u(x) = 1 ) and ( v(x) = \sec(x) - \tan(x) ).

Using the quotient rule: [ \frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} ]

Now, find the derivatives of ( u(x) ) and ( v(x) ): [ u'(x) = 0 ] [ v'(x) = \sec(x)\tan(x) - \sec^2(x) ]

Plug these derivatives into the quotient rule: [ \frac{d}{dx}\left(\frac{1}{\sec(x) - \tan(x)}\right) = \frac{0 \cdot (\sec(x) - \tan(x)) - 1 \cdot (\sec(x)\tan(x) - \sec^2(x))}{(\sec(x) - \tan(x))^2} ]

Simplify: [ \frac{-\sec(x)\tan(x) + \sec^2(x)}{(\sec(x) - \tan(x))^2} ]

This is the derivative of ( \frac{1}{\sec(x) - \tan(x)} ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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