If #f(x)= cot5 x # and #g(x) = 2x^2 -1 #, how do you differentiate #f(g(x)) # using the chain rule?

Answer 1

#(df)/(dx)-4xcsc^2(10x^2-5)#

If #f(x)=cot5x# and #g(x)=2x^2-1#
#f(g(x))=cot(5*(2x^2-1))#
As according to chain rule, #(df)/(dx)=(df)/(dg)xx(dg)/(dx)#
#(df)/(dx)=-csc^2(5*(2x^2-1))xxd/(dx)(2x^2-1)#
= #-csc^2(10x^2-5)xx4x)#
= #-4xcsc^2(10x^2-5)#
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Answer 2

To differentiate ( f(g(x)) ) using the chain rule, follow these steps:

  1. Compute the derivative of the outer function ( f(x) ) with respect to its inner function ( g(x) ). This is denoted as ( f'(g(x)) ).

  2. Compute the derivative of the inner function ( g(x) ) with respect to ( x ), denoted as ( g'(x) ).

  3. Multiply ( f'(g(x)) ) by ( g'(x) ).

  4. Substitute ( g(x) ) back into the resulting expression.

Applying this to ( f(g(x)) = \cot^5(g(x)) ) and ( g(x) = 2x^2 - 1 ):

  1. Compute ( f'(g(x)) ) by finding the derivative of ( \cot^5(x) ), which is ( -5\cot^4(x)\csc^2(x) ).

  2. Compute ( g'(x) ) by finding the derivative of ( 2x^2 - 1 ), which is ( 4x ).

  3. Multiply ( f'(g(x)) ) by ( g'(x) ) to get ( -20x\cot^4(2x^2 - 1)\csc^2(2x^2 - 1) ).

Therefore, the derivative of ( f(g(x)) ) with respect to ( x ) using the chain rule is ( -20x\cot^4(2x^2 - 1)\csc^2(2x^2 - 1) ).

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Answer from HIX Tutor

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.

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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