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Algebra
Question

Find the derivatives of the function \( f \) for \...

Find the derivatives of the function \( f \) for \( n = 1,2,3 \) , and \( 4 . \) 

\( f ( x ) = x ^ { n } \sin x \) 

\( n = 1 f ^ { \prime } ( x ) = x \cos ( x ) + \sin ( x ) \) 

\( n = 2 f ^ { \prime } ( x ) = x ^ { 2 } \cos ( x ) + 2 x \sin ( x ) \) 

\( n = 3 f ^ { \prime } ( x ) = x ^ { 3 } \cos ( x ) + 3 x ^ { 2 } \sin ( x ) \) 

\( n = 4 f ^ { \prime } ( x ) = x ^ { 4 } \cos ( x ) + 4 x ^ { 3 } \sin ( x ) \) 

Use the results to write a general rule for \( f ^ { \prime } ( x ) \) in terms of \( n . \) 

\( f ^ { \prime } ( x ) = x ^ { n } \cos ( x ) + n k ^ { n - 1 } \sin ( x )\) 

Answer

f'(x) = \(x^{n}\) Cos( x ) + nx\(^{n- 1}\)Sin(x)

Solution
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