How do you differentiate #f(x) = (sinx)/(1cosx)# using the quotient rule?
For derivatives, the Quotient Rule stipulates:
then
See also the Quotient Rule article at https://tutor.hix.ai
so
Using the Quotient Rule as a guide, let's put everything together and then simplify:
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To differentiate ( f(x) = \frac{\sin x}{1  \cos x} ) using the quotient rule, follow these steps:

Apply the quotient rule, which states that for functions ( u(x) ) and ( v(x) ), the derivative of ( \frac{u(x)}{v(x)} ) is given by ( \frac{u'(x)v(x)  u(x)v'(x)}{(v(x))^2} ).

Identify ( u(x) = \sin x ) and ( v(x) = 1  \cos x ).

Compute the derivatives ( u'(x) ) and ( v'(x) ), which are ( u'(x) = \cos x ) and ( v'(x) = \sin x ).

Apply the quotient rule formula: [ f'(x) = \frac{(u'(x)v(x)  u(x)v'(x))}{(v(x))^2} ] [ = \frac{(\cos x)(1  \cos x)  (\sin x)(\sin x)}{(1  \cos x)^2} ]

Simplify the expression to obtain the derivative ( f'(x) ).
[ f'(x) = \frac{\cos x  \cos^2 x + \sin^2 x}{(1  \cos x)^2} ]
[ = \frac{\cos x  \cos^2 x + (1  \cos^2 x)}{(1  \cos x)^2} ]
[ = \frac{\cos x + 1  2\cos^2 x}{(1  \cos x)^2} ]
[ = \frac{\cos x + 1  2(1  \sin^2 x)}{(1  \cos x)^2} ]
[ = \frac{\cos x + 1  2 + 2\sin^2 x}{(1  \cos x)^2} ]
[ = \frac{\sin^2 x + \cos x  1}{(1  \cos x)^2} ]
[ = \frac{\sin^2 x + \cos x  1}{(1  \cos x)^2} ]
[ = \frac{\sin^2 x + \cos x  1}{(1  \cos x)^2} ]
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To differentiate ( f(x) = \frac{\sin(x)}{1  \cos(x)} ) using the quotient rule:
 Apply the quotient rule formula, ( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot u'  u \cdot v'}{v^2} ).
 Identify ( u = \sin(x) ) and ( v = 1  \cos(x) ).
 Compute the derivatives ( u' ) and ( v' ):
 ( u' = \cos(x) ) (derivative of sine function)
 ( v' = \sin(x) ) (derivative of ( 1  \cos(x) ))
 Substitute ( u, v, u', ) and ( v' ) into the quotient rule formula.
 Simplify the expression.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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