How do you differentiate # f(x)=x^7csc(x)# using the product rule?

Answer 1

#f'(x)= -x^6csc(x)[xcot(x)-7]#

Keep the first factor and find the derivative of the second then add that to the derivative of the first factor and keep the 2nd factor untouched

#f(x)=uv#

#f'(x)=uv'+u'v#

#f(x)= x^7csc(x)#

#f'(x)=x^7(-csc(x) cot(x))+7x^6csc(x)#

Factor out #-x^6csc(x)#

#f'(x)= -x^6csc(x)[xcot(x)-7]#

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Answer 2

Charles Arocha

To differentiate ( f(x) = x^7 \csc(x) ) using the product rule:

Identify the two functions being multiplied: ( u(x) = x^7 ) and ( v(x) = \csc(x) ).
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) = 7x^6 )
- ( v'(x) = -\csc(x) \cot(x) )
Substitute the derivatives into the product rule formula:
- ( f'(x) = (7x^6)(\csc(x)) + (x^7)(-\csc(x) \cot(x)) )
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.