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How do you differentiate #f(x)=cscx# using the quotient rule?

Answer 1

Charles Arocha

First we must write #f(x)# as a quotient.

#f(x) = cscx = 1/sinx#

Now use the quotient rule to differentiate:

#f'(x) = ((0)(sinx)-(1)(cosx))/(sinx)^2#

# = -cosx/sin^2x#

# = -cosx/sinx 1/sinx#

# = -cotx cscx = -cscx cotx#

(Choose a way of expressing the answer that you (or your grader) prefer.)

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Answer 2

Paisley Johnson

To differentiate ( f(x) = \csc(x) ) using the quotient rule, you first express it as ( f(x) = \frac{1}{\sin(x)} ). Then apply the quotient rule:

[ f'(x) = \frac{d}{dx}\left(\frac{1}{\sin(x)}\right) = \frac{(\sin(x))' \cdot 1 - \sin(x) \cdot (1)'}{(\sin(x))^2} ]

[ = \frac{\cos(x) \cdot 1 - \sin(x) \cdot 0}{\sin^2(x)} = \frac{\cos(x)}{\sin^2(x)} = \csc(x) \cot(x) ]

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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.