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

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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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