How do you differentiate #f(x)=ln(sinx)/cosx# using the quotient rule?
The quotient rule states that
Find each derivative separately:
To differentiate the natural logarithm function, use the chain rule, which for a natural logarithm function states that
As for the other derivative,
Plugging these both back in, we see that
This can be written as
As with many trigonometric functions, this can be rewritten in many ways, including
By signing up, you agree to our Terms of Service and Privacy Policy
To differentiate ( f(x) = \frac{\ln(\sin x)}{\cos x} ) using the quotient rule:
 Apply the quotient rule, which states that for a function ( \frac{u}{v} ), the derivative is given by ( \frac{u'v  uv'}{v^2} ).
 Let ( u = \ln(\sin x) ) and ( v = \cos x ).
 Compute ( u' ) and ( v' ) as the derivatives of ( u ) and ( v ) with respect to ( x ), respectively.
 Substitute ( u' ), ( v' ), ( u ), and ( v ) into the quotient rule formula.
 Simplify the expression.
Therefore, the derivative ( f'(x) ) of ( f(x) ) with respect to ( x ) is given by:
[ f'(x) = \frac{\frac{d}{dx}(\ln(\sin x))\cdot \cos x  \ln(\sin x) \cdot \frac{d}{dx}(\cos x)}{\cos^2 x} ]
Then, compute the derivatives of ( \ln(\sin x) ) and ( \cos x ) with respect to ( x ) using the chain rule and simplify the expression.
By signing up, you agree to our Terms of Service and Privacy Policy
To differentiate ( f(x) = \frac{\ln(\sin(x))}{\cos(x)} ) using the quotient rule:

Identify ( u ) and ( v ) as the numerator and denominator, respectively. ( u = \ln(\sin(x)) ) and ( v = \cos(x) ).

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

Find the derivatives of ( u ) and ( v ): [ u' = \frac{1}{\sin(x)} \cdot \cos(x) = \frac{\cos(x)}{\sin(x)} ] [ v' = \sin(x) ]

Substitute ( u ), ( v ), ( u' ), and ( v' ) into the quotient rule formula: [ f'(x) = \frac{\frac{\cos(x)}{\sin(x)}\cos(x)  \ln(\sin(x))(\sin(x))}{\cos^2(x)} ]

Simplify the expression: [ f'(x) = \frac{\cos^2(x) + \ln(\sin(x))\sin(x)}{\sin(x)\cos^2(x)} ]
By signing up, you agree to our Terms of Service and Privacy Policy
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.
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7