# If #f(x) =-e^(x) # and #g(x) = 3csc^2x^2 #, what is #f'(g(x)) #?

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To find ( f'(g(x)) ), first, substitute ( g(x) ) into ( f(x) ) to get ( f(g(x)) ). Then find the derivative of ( f(g(x)) ) with respect to ( x ).

Given: [ f(x) = -e^x ] [ g(x) = 3 \csc^2(x^2) ]

Substituting ( g(x) ) into ( f(x) ): [ f(g(x)) = -e^{3 \csc^2(x^2)} ]

Now, find the derivative of ( f(g(x)) ): [ f'(g(x)) = -e^{3 \csc^2(x^2)} \cdot \frac{d}{dx} (3 \csc^2(x^2)) ]

Apply chain rule to find ( \frac{d}{dx} (3 \csc^2(x^2)) ): [ \frac{d}{dx} (3 \csc^2(x^2)) = -6 \cot(x^2) \csc(x^2) \csc^2(x^2) ]

Therefore, ( f'(g(x)) = -e^{3 \csc^2(x^2)} \cdot (-6 \cot(x^2) \csc(x^2) \csc^2(x^2)) ).

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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.

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.

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.

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