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How do you use the chain rule to differentiate #y=cos(2x+3)#?

Answer 1

Charles Arocha

#y'=-2sin(2x+3)#

#y'=-sin(2x+3)*2#

#=-2sin(2x+3)#

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Answer 2

Cameron Alexander

To differentiate ( y = \cos(2x + 3) ) using the chain rule:

Identify the outer function, which is ( \cos(u) ), where ( u = 2x + 3 ).
Differentiate the outer function with respect to its inner variable ( u ), which is ( \frac{d}{du} \cos(u) = -\sin(u) ).
Differentiate the inner function with respect to the variable ( x ), which is ( \frac{d}{dx} (2x + 3) = 2 ).
Apply the chain rule by multiplying the derivatives from steps 2 and 3: ( \frac{dy}{dx} = -\sin(u) \cdot 2 ).
Replace ( u ) with ( 2x + 3 ) to get the final derivative: ( \frac{dy}{dx} = -2\sin(2x + 3) ).

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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.