How do you differentiate #f(x)=(cot(x))^2 # using the chain rule?
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To differentiate ( f(x) = (\cot(x))^2 ) using the chain rule, you first find the derivative of the outer function and then multiply it by the derivative of the inner function.
The derivative of ( (\cot(x))^2 ) is ( 2 \cot(x) \cdot (\csc^2(x)) ).
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To differentiate ( f(x) = (\cot(x))^2 ) using the chain rule, follow these steps:

Recognize that ( (\cot(x))^2 ) is a composition of functions, with the outer function being the square function ( g(u) = u^2 ) and the inner function being ( u = \cot(x) ).

Differentiate the outer function with respect to its variable. In this case, differentiate ( g(u) = u^2 ) with respect to ( u ) to get ( g'(u) = 2u ).

Differentiate the inner function with respect to the independent variable, ( x ). In this case, differentiate ( u = \cot(x) ) with respect to ( x ) using the derivative of cotangent formula, which is ( \frac{d}{dx} \cot(x) = \csc^2(x) ).

Apply the chain rule, which states that if ( y = g(u) ) and ( u = h(x) ), then ( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} ).

Substitute the derivatives obtained in steps 2 and 3 into the chain rule formula to find ( \frac{df}{dx} ), the derivative of ( f(x) = (\cot(x))^2 ) with respect to ( x ).

Simplify the expression if necessary.
This process will yield the derivative of ( f(x) = (\cot(x))^2 ) 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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