How do you calculate, the derivative of #cos^3x#?

Answer 1

I would use the Chain Rule to deal separately with the #()^3# and #cos# functions getting: #y'=3cos^2(x)(-sin(x))=-3sin(x)cos^2(x)#

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Answer 2

Adam Adkins

To calculate the derivative of ( \cos^3(x) ), you can use the chain rule and the power rule. The chain rule states that if ( u ) is a function of ( x ) and ( v ) is a function of ( u ), then the derivative of ( v ) with respect to ( x ) is ( \frac{dv}{dx} = \frac{dv}{du} \cdot \frac{du}{dx} ). Applying this to ( \cos^3(x) ), let ( u = \cos(x) ), then ( \frac{du}{dx} = -\sin(x) ). Now differentiate ( u^3 ) with respect to ( u ) to get ( \frac{d}{du}(u^3) = 3u^2 ). Finally, combine the derivatives using the chain rule: ( \frac{d}{dx}(\cos^3(x)) = 3(\cos(x))^2 \cdot (-\sin(x)) ). So, the derivative of ( \cos^3(x) ) is ( -3\cos^2(x)\sin(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.