How do you differentiate #f(x)=sec^2x-tan^2x#?

Answer 1

See below

#(df)/dx=2secx(dsecx)/dx-2tanx(dtanx)/dx=2secxtanxsecx-2tanxsec^2x=2sec^2xtanx-2sec^2xtanx=0#

This obvious result is the consecuence of the fact

#sec^2x-tan^2x=1/(cos^x)-sin^2x/cos^2x=(1-sin^2x)/cos^2x=cos^2x/cos^2x=1#
Therefore #f(x)# is a constant function for every #x#. For this reason his derivative is zero
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Answer 2

To differentiate ( f(x) = \sec^2(x) - \tan^2(x) ), you can use the following steps:

  1. Identify the individual functions: ( \sec(x) ) and ( \tan(x) ).
  2. Apply the chain rule to differentiate each function.
  3. Substitute the derivatives back into the original expression and simplify.

Differentiate ( \sec(x) ) and ( \tan(x) ):

( \frac{d}{dx}(\sec(x)) = \sec(x) \tan(x) ) ( \frac{d}{dx}(\tan(x)) = \sec^2(x) )

Substitute derivatives back into the original expression:

( f'(x) = \sec(x) \tan(x) - \sec^2(x) )

Simplify by factoring out ( \sec(x) ):

( f'(x) = \sec(x)(\tan(x) - \sec(x)) )

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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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