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Algebra
Question

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 $$n .$$

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

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