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How do you differentiate #f(x)=csc4x * sin7x# using the product rule?

Answer 1

Charles Arocha

#f'(x)=7csc4x*cos7x-4sin7x*csc4x*cot4x#

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#f(x)=csc4x * sin7x#

#f'(x)=[csc4x*7cos(7x)]+[-4csc4x*cot4x*sin7x]#

#f'(x)=7csc4x*cos7x-4sin7x*csc4x*cot4x#

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Answer 2

Samantha Adkison

To differentiate ( f(x) = \csc^4(x) \times \sin(7x) ) using the product rule, follow these steps:

Let ( u(x) = \csc^4(x) ) and ( v(x) = \sin(7x) ).
Find the derivatives of ( u(x) ) and ( v(x) ).
Apply the product rule: ( (uv)' = u'v + uv' ).
Substitute the derivatives and functions into the product rule formula.
Simplify the result.

The derivatives are: [ u'(x) = -4\csc^3(x)\cot(x) ] [ v'(x) = 7\cos(7x) ]

Applying the product rule: [ (uv)' = u'v + uv' ]

Substituting the derivatives and functions: [ (f(x))' = (-4\csc^3(x)\cot(x)) \times \sin(7x) + \csc^4(x) \times 7\cos(7x) ]

Simplify the expression if necessary.

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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.