# How do you find the derivative of #2sec((3x+1)^(1/2))#?

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To find the derivative of ( 2\sec\left((3x+1)^{\frac{1}{2}}\right) ), you would use the chain rule. The derivative can be found using the formula:

[ \frac{d}{dx}\left[f(g(x))\right] = f'(g(x)) \cdot g'(x) ]

Where ( f(x) = 2\sec(x) ) and ( g(x) = (3x+1)^{\frac{1}{2}} ).

First, find the derivative of ( f(x) = 2\sec(x) ) which is ( f'(x) = 2\sec(x)\tan(x) ).

Next, find the derivative of ( g(x) = (3x+1)^{\frac{1}{2}} ) using the power rule, ( g'(x) = \frac{1}{2}(3x+1)^{-\frac{1}{2}} \cdot 3 ).

Now, apply the chain rule:

[ \frac{d}{dx}\left[2\sec\left((3x+1)^{\frac{1}{2}}\right)\right] = 2\sec\left((3x+1)^{\frac{1}{2}}\right)\tan\left((3x+1)^{\frac{1}{2}}\right) \cdot \frac{1}{2}(3x+1)^{-\frac{1}{2}} \cdot 3 ]

Combining terms, the derivative is:

[ \frac{3\tan\left((3x+1)^{\frac{1}{2}}\right)}{(3x+1)^{\frac{1}{2}}} ]

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To find the derivative of (2\sec(\sqrt{3x+1})), you can use the chain rule. First, let (u = \sqrt{3x+1}). Then, the derivative of (u) with respect to (x) is (\frac{du}{dx} = \frac{1}{2\sqrt{3x+1}}). Now, apply the chain rule:

[\frac{d}{dx}\left(2\sec(\sqrt{3x+1})\right) = 2\sec(\sqrt{3x+1})\tan(\sqrt{3x+1})\cdot \frac{du}{dx}]

Substitute the expression for (\frac{du}{dx}) into the equation:

[\frac{d}{dx}\left(2\sec(\sqrt{3x+1})\right) = 2\sec(\sqrt{3x+1})\tan(\sqrt{3x+1})\cdot \frac{1}{2\sqrt{3x+1}}]

Now, simplify:

[\frac{d}{dx}\left(2\sec(\sqrt{3x+1})\right) = \frac{\sec(\sqrt{3x+1})\tan(\sqrt{3x+1})}{\sqrt{3x+1}}]

So, the derivative of (2\sec(\sqrt{3x+1})) with respect to (x) is (\frac{\sec(\sqrt{3x+1})\tan(\sqrt{3x+1})}{\sqrt{3x+1}}).

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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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