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How do you differentiate #f(x)=sec(e^(x)-3x ) # using the chain rule?

Answer 1

Christopher Agresta

#f'(x)=(e^x-3)sec(e^x-3x)tan(e^x-3x)#

#f(x)=sec(e^x-3x)#

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Answer 2

Emma Aldridge

To differentiate ( f(x) = \sec(e^x - 3x) ) using the chain rule, follow these steps:

Identify the outer function as ( \sec(u) ) and the inner function as ( u = e^x - 3x ).
Differentiate the inner function ( u ) with respect to ( x ) to find ( \frac{du}{dx} ).
Apply the chain rule: [ \frac{d}{dx}[\sec(u)] = \sec(u) \tan(u) \frac{du}{dx} ]
Substitute ( u = e^x - 3x ) into the expression.
Compute ( \frac{du}{dx} ).
Substitute all values into the chain rule formula to find ( \frac{d}{dx}[f(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.