If #f(x)= csc 7 x # and #g(x) = 3x^2 5 #, how do you differentiate #f(g(x)) # using the chain rule?
First, note the chain rule states that
In the case of a cosecant function, the chain rule states that
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To differentiate ( f(g(x)) ) using the chain rule, follow these steps:
 Find ( f'(x) ) and ( g'(x) ).
 Substitute ( g(x) ) into ( f(x) ) to get ( f(g(x)) ).
 Differentiate ( f(g(x)) ) with respect to ( x ) using the chain rule, which states that ( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) ).
Now, let's apply these steps:

Find ( f'(x) ) and ( g'(x) ):
 ( f'(x) = \csc(x) \cot(x) )
 ( g'(x) = 6x )

Substitute ( g(x) ) into ( f(x) ):
 ( f(g(x)) = \csc(3x^2  5) )

Apply the chain rule:
 ( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) )
 ( = \csc(3x^2  5) \cot(3x^2  5) \cdot 6x )
So, the derivative of ( f(g(x)) ) with respect to ( x ) is ( 6x\csc(3x^2  5) \cot(3x^2  5) ).
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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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