How do you differentiate #f(x) = 4/sqrt(tan^3(1/x) # using the chain rule?
By rewriting a bit,
By applying Power Rule & Chain Rule repeatedly,
By cleaning up a bit,
#=(6tan^2(1/x)sec^2(1/x))/(x^2sqrt(tan^9(1/x))) =(6tan^2(1/x)sec^2(1/x))/(x^2tan^2(1/x)sqrt(tan^5(1/x))) =(6sec^2(1/x))/(x^2sqrt(tan^5(1/x)))#
I hope that this was clear.
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To differentiate the function ( f(x) = \frac{4}{\sqrt{\tan^3(1/x)}} ) using the chain rule, follow these steps:

Identify the outer function ( u ) and the inner function ( v ).
 Let ( u = \frac{4}{\sqrt{v}} ), where ( v = \tan^3(1/x) ).

Differentiate the outer function with respect to ( u ), denoted as ( \frac{du}{dv} ).
 ( \frac{du}{dv} = \frac{1}{2}v^{\frac{3}{2}} ).

Differentiate the inner function with respect to ( x ), denoted as ( \frac{dv}{dx} ).
 Apply the chain rule for differentiation: ( \frac{dv}{dx} = \frac{d}{dx}(\tan^3(1/x)) ).
 Use the chain rule for the derivative of ( \tan^3(u) ), where ( u = 1/x ): ( \frac{dv}{dx} = 3\tan^2(1/x)\cdot\frac{d}{dx}(1/x) ).
 Compute ( \frac{d}{dx}(1/x) = \frac{1}{x^2} ).

Combine the derivatives using the chain rule.
 ( \frac{df}{dx} = \frac{du}{dv} \cdot \frac{dv}{dx} = \frac{1}{2}v^{\frac{3}{2}} \cdot 3\tan^2(1/x) \cdot (\frac{1}{x^2}) ).

Substitute ( v = \tan^3(1/x) ) back into the equation.
 ( \frac{df}{dx} = \frac{3}{2}\frac{\tan^2(1/x)}{x^2\sqrt{\tan^3(1/x)}} ).
This is the derivative of the function ( f(x) ) with respect to ( x ) using the chain rule.
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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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