How do you differentiate #f(x)=-cosx*(-2sinx)# using the product rule?

Answer 1

Find the derivative of the first term and multiply it with the second term, add that to the derivative of the second term multiplied with the first term.

So # dy/dx -cos(x)*-2sin(x) = (-cos(x))'(-2sin(x)) + (-cos(x))(-2sin(x))#

#= sin(x)*(-2sin(x)) + (-cos(x))(-2cos(x)) #

#= -2sin^2(x) + 2cos^2(x)#

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Answer 2

William Abdalla

To differentiate ( f(x) = -\cos(x) \cdot (-2\sin(x)) ) using the product rule, you can follow these steps:

Identify the two functions being multiplied: ( u(x) = -\cos(x) ) and ( v(x) = -2\sin(x) ).
Apply the product rule: ( f'(x) = u'(x)v(x) + u(x)v'(x) ).
Differentiate each function: ( u'(x) = \sin(x) ) and ( v'(x) = -2\cos(x) ).
Substitute the derivatives and original functions into the product rule formula.
Simplify the expression to get the final result.

( f'(x) = \sin(x) \cdot (-2\sin(x)) + (-\cos(x)) \cdot (-2\cos(x)) ) ( f'(x) = -2\sin^2(x) + 2\cos^2(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.