If #f(x)= tan8 x # and #g(x) = e^(5x ) #, how do you differentiate #f(g(x)) # using the chain rule?

Answer 1

#f(g(x))=40e^(5x)sec^2(8e^(5x))#

As #f(x)=tan8x# and #g(x)=e^(5x)#
#f(g(x))=tan(8e^(5x))#
and #(df)/(dx)=(d(tan8e^(5x)))/(d(8e^(5x)))*(d(8e^(5x)))/(dx)#
= #sec^2(8e^(5x))*8*e^(5x)*5#
= #40e^(5x)sec^2(8e^(5x))#
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Answer 2

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

  1. Find the derivative of ( g(x) ) with respect to ( x ), denoted as ( g'(x) ).
  2. Substitute ( g(x) ) into ( f(x) ) to get ( f(g(x)) ).
  3. Differentiate ( f(g(x)) ) with respect to ( x ).
  4. Apply the chain rule by multiplying the derivative of the outer function ( f ) with the derivative of the inner function ( g(x) ), i.e., ( f'(g(x)) \cdot g'(x) ).

So, to find ( \frac{d}{dx}[f(g(x))] ):

  1. Find ( g'(x) ) by differentiating ( g(x) ). ( g'(x) = 5e^{5x} )

  2. Substitute ( g(x) ) into ( f(x) ) to get ( f(g(x)) ). ( f(g(x)) = \tan^8(g(x)) )

  3. Differentiate ( f(g(x)) ) with respect to ( x ). ( \frac{d}{dx}[f(g(x))] = 8\tan^7(g(x)) \cdot \sec^2(g(x)) \cdot g'(x) )

  4. Finally, substitute ( g'(x) ) into the expression derived in step 3 to get the final result. ( \frac{d}{dx}[f(g(x))] = 8\tan^7(g(x)) \cdot \sec^2(g(x)) \cdot 5e^{5x} )

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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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