How do you differentiate # g(x) = (cotx + cscx)/(tanx  sinx) #?
= #([tanxcsc^2xcscx+cscx+cotx secxcscxsec^2xcscx+cosxcotx+cotx])/(tanxsinx)^2#
= #([tanxcsc^2x+2cotx secxcscxsec^2xcscx+cosxcotx])/(tanxsinx)^2#
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To differentiate ( g(x) = \frac{\cot(x) + \csc(x)}{\tan(x)  \sin(x)} ), follow these steps:
 Express all trigonometric functions in terms of sine and cosine.
 Apply the quotient rule for differentiation.
 Simplify the expression.
Here are the detailed steps:

Rewrite cotangent, cosecant, and tangent functions in terms of sine and cosine:
 ( \cot(x) = \frac{\cos(x)}{\sin(x)} )
 ( \csc(x) = \frac{1}{\sin(x)} )
 ( \tan(x) = \frac{\sin(x)}{\cos(x)} )

Apply the quotient rule:
 Let ( u(x) = \cot(x) + \csc(x) ) and ( v(x) = \tan(x)  \sin(x) ).
 Then, ( g(x) = \frac{u(x)}{v(x)} ).
 Differentiate ( u(x) ) and ( v(x) ) separately.
 Use the quotient rule: ( \frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{v(x)u'(x)  u(x)v'(x)}{(v(x))^2} ).

Simplify the expression:
 Substitute the derivatives of ( u(x) ) and ( v(x) ) into the quotient rule expression.
 Simplify the resulting expression.
The final expression after differentiation and simplification would be the derivative of ( g(x) ).
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To differentiate the function g(x) = (cot(x) + csc(x))/(tan(x)  sin(x)), you can use the quotient rule:
Let u = cot(x) + csc(x) and v = tan(x)  sin(x).
Then, g'(x) = (u'v  uv') / v^2, where u' and v' are the derivatives of u and v respectively.
Now, find the derivatives: u' = csc^2(x)  csc(x)cot(x) v' = sec^2(x)  cos(x)
Plug these values into the quotient rule formula:
g'(x) = ((csc^2(x)  csc(x)cot(x))(tan(x)  sin(x))  (cot(x) + csc(x))(sec^2(x)  cos(x))) / (tan(x)  sin(x))^2
Simplify the expression if necessary.
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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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