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How do you find the derivative of #y = 4 / cosx#?

Answer 1

Ezekiel Alexander

#4 sec x tan x#

#( 4 / cosx)' = 4 (( cosx)^{-1} )'#

#= 4 (-1( cosx)^{-2} (-sin x))#

#= 4 (1/( cosx)^{2} (sin x))#

#= 4 sec x tan x#

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Answer 2

Isaiah Allen

To find the derivative of ( y = \frac{4}{\cos(x)} ), you can use the quotient rule. The quotient rule states that if you have a function in the form ( \frac{f(x)}{g(x)} ), then the derivative is given by:

[ \left(\frac{f(x)}{g(x)}\right)' = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{(g(x))^2} ]

For ( y = \frac{4}{\cos(x)} ), ( f(x) = 4 ) and ( g(x) = \cos(x) ). The derivatives of ( f(x) ) and ( g(x) ) are ( f'(x) = 0 ) and ( g'(x) = -\sin(x) ), respectively.

Substituting into the quotient rule formula:

[ \begin{align*} y' &= \frac{(0)(\cos(x)) - (4)(-\sin(x))}{(\cos(x))^2} \ &= \frac{4\sin(x)}{\cos^2(x)} \ &= 4 \cdot \frac{\sin(x)}{\cos^2(x)} \ &= 4 \cdot \tan(x) \end{align*} ]

So, ( y' = 4 \tan(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.