How do you find the derivative of #1/(sec x - tan x)#?

Answer 1

#1/(1-sinx)#

First, simplify the function:

#y=1/(1/cosx-sinx/cosx)=1/((1-sinx)/cosx)=cosx/(1-sinx)#

From here, use the quotient rule.

#y^'=(d/dx(cosx)*(1-sinx)-(cosx)*d/dx(1-sinx))/(1-sinx)^2#
#y^'=((-sinx)(1-sinx)-cosx(-cosx))/(1-sinx)^2#
#y^'=(-sinx+sin^2x+cos^2x)/(1-sinx)^2#
Recall that #sin^2x+cos^2x=1#:
#y^'=(1-sinx)/(1-sinx)^2#
#y^'=1/(1-sinx)#

Note that there are many different ways this can be written, but this is the simplest.

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Answer 2

To find the derivative of ( \frac{1}{\sec x - \tan x} ), use the quotient rule. The quotient rule states that if you have a function of the form ( \frac{u(x)}{v(x)} ), the derivative is given by:

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

Where ( u(x) ) and ( v(x) ) are differentiable functions of ( x ). Applying this rule, the derivative of ( \frac{1}{\sec x - \tan x} ) is:

[ \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} ]

[ = \frac{-\sec x \tan x - \sec^2 x}{(\sec x - \tan x)^2} ]

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

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Answer from HIX Tutor

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.

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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