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How do you differentiate #g(x) =e^(1-x^2)sec# using the product rule?

Answer 1

Ezekiel Alexander

I assume this should be #g(x) =e^(1-x^2)secx#

Take note of that

#d/dx(e^(1-x^2))# may be found using thechain rule,

#d/dx(e^(1-x^2)) = -2xe^(1-x^2)#

And #d/dx(secx) = secxtanx#, so

#d/dx(e^(1-x^2)) = -2xe^(1-x^2)secx+e^(1-x^2)secx tanx#

To taste, factor and rewrite.

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Answer 2

Christopher Agresta

To differentiate ( g(x) = e^{1-x^2} \sec(x) ) using the product rule, you would follow these steps:

Identify the two functions being multiplied: ( f(x) = e^{1-x^2} ) and ( h(x) = \sec(x) ).
Apply the product rule: ( g'(x) = f'(x)h(x) + f(x)h'(x) ).
Differentiate ( f(x) = e^{1-x^2} ) with respect to ( x ) to find ( f'(x) ).
Differentiate ( h(x) = \sec(x) ) with respect to ( x ) to find ( h'(x) ).
Substitute ( f'(x) ), ( f(x) ), ( h'(x) ), and ( h(x) ) into the product rule equation and simplify.

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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.