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How do you differentiate #f(x)=e^(-6x) sin(3x)# using the product rule?

Answer 1

Ezra Adams

#f'(x)=3e^{-6x}[cos(3x)-2sin(3x)]#

#f'(x)=frac{d}{dx}(e^{-6x}sin(3x))#

#=e^{-6x}frac{d}{dx}(sin(3x))+sin(3x)frac{d}{dx}(e^{-6x})#

#=e^{-6x}(3cos(3x))+sin(3x)(-6e^{-6x})#

#=3e^{-6x}[cos(3x)-2sin(3x)]#

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Answer 2

Ezra Adams

To differentiate the function ( f(x) = e^{-6x} \sin(3x) ) using the product rule:

Identify the two functions: ( u(x) = e^{-6x} ) and ( v(x) = \sin(3x) ).
Apply the product rule: ( f'(x) = u'(x)v(x) + u(x)v'(x) ).
Find the derivatives of ( u(x) ) and ( v(x) ): ( u'(x) = -6e^{-6x} ) and ( v'(x) = 3\cos(3x) ).
Substitute the derivatives into the product rule formula: ( f'(x) = (-6e^{-6x})\sin(3x) + e^{-6x}(3\cos(3x)) ).
Simplify the expression: ( f'(x) = -6e^{-6x}\sin(3x) + 3e^{-6x}\cos(3x) ).

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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.