If #f(x) =tan^2(x/2) # and #g(x) = sqrt(5x1 #, what is #f'(g(x)) #?
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To find ( f'(g(x)) ), we first need to find ( f'(x) ) and ( g'(x) ), and then substitute ( g(x) ) into ( f'(x) ).
Given ( f(x) = \tan^2(x/2) ) and ( g(x) = \sqrt{5x  1} ):

Find ( f'(x) ): ( f(x) = \tan^2(x/2) ) ( f'(x) = \frac{d}{dx}[\tan^2(x/2)] ) ( f'(x) = 2\tan(x/2) \sec^2(x/2) )

Find ( g'(x) ): ( g(x) = \sqrt{5x  1} ) ( g'(x) = \frac{d}{dx}[\sqrt{5x  1}] ) ( g'(x) = \frac{1}{2\sqrt{5x  1}} \cdot 5 ) ( g'(x) = \frac{5}{2\sqrt{5x  1}} )

Substitute ( g(x) ) into ( f'(x) ): ( f'(g(x)) = 2\tan(g(x)/2) \sec^2(g(x)/2) ) ( f'(g(x)) = 2\tan\left(\frac{\sqrt{5x  1}}{2}\right) \sec^2\left(\frac{\sqrt{5x  1}}{2}\right) )
Therefore, ( f'(g(x)) = 2\tan\left(\frac{\sqrt{5x  1}}{2}\right) \sec^2\left(\frac{\sqrt{5x  1}}{2}\right) ).
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[ f'(g(x)) = \frac{f'(u)}{g'(x)} ]
Where ( u = g(x) )
[ f(u) = \tan^2\left(\frac{u}{2}\right) ] [ f'(u) = 2\tan\left(\frac{u}{2}\right)\sec^2\left(\frac{u}{2}\right) ]
[ g(x) = \sqrt{5x  1} ] [ g'(x) = \frac{5}{2\sqrt{5x  1}} ]
[ f'(g(x)) = \frac{2\tan\left(\frac{\sqrt{5x  1}}{2}\right)\sec^2\left(\frac{\sqrt{5x  1}}{2}\right)}{\frac{5}{2\sqrt{5x  1}}} ]
[ f'(g(x)) = \frac{4\tan\left(\frac{\sqrt{5x  1}}{2}\right)\sec^2\left(\frac{\sqrt{5x  1}}{2}\right)}{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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