# If #f(x) =sec^3(x/2) # and #g(x) = sqrt(2x-1 #, what is #f'(g(x)) #?

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the Chain Rule tells

Thus:

Alternatively, once you get practice, more simply:

Given:

thus you can apply the rule ricorsively.

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To find ( f'(g(x)) ), you first need to find the derivative of ( f(x) ) with respect to ( x ), and then substitute ( g(x) ) into the derivative of ( f(x) ).

Given ( f(x) = \sec^3\left(\frac{x}{2}\right) ) and ( g(x) = \sqrt{2x - 1} ), the derivative ( f'(x) ) of ( f(x) ) with respect to ( x ) is:

[ f'(x) = \frac{d}{dx}\left[\sec^3\left(\frac{x}{2}\right)\right] ]

Applying the chain rule, ( f'(x) ) can be found as follows:

[ f'(x) = 3\sec^2\left(\frac{x}{2}\right) \cdot \sec\left(\frac{x}{2}\right) \tan\left(\frac{x}{2}\right) \cdot \frac{1}{2} ]

Now, substitute ( g(x) = \sqrt{2x - 1} ) into ( f'(x) ) to get ( f'(g(x)) ):

[ f'(g(x)) = 3\sec^2\left(\frac{\sqrt{2x - 1}}{2}\right) \cdot \sec\left(\frac{\sqrt{2x - 1}}{2}\right) \tan\left(\frac{\sqrt{2x - 1}}{2}\right) \cdot \frac{1}{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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