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

Question

f(x) = xⁿ sin x

I have done that and then it asks for:

Use the results to write a general rule for f'(x)

 in terms of n.



Answer 1

I presume your results are something in line with:

ƒ'(x) = x cos( x ) + sin( x )

ƒ'(x) = x² cos( x ) + 2x sin( x )

ƒ'(x) = x³ cos( x ) + 3x² sin( x )

If so, then this is what you can do

You notice that due to the Product Rule of derivatives, xⁿ seems to "survive" the differentiation and stick around on the first term, namely being cos( x ), the derivative of sin( x ).

Also, consequently from the Product Rule of derivatives, sin( x ) "survives" the differentiation and sticks to the secondary term. Following how Product Rule works,

( u(x) * v(x) )' = u(x) * v'(x) + u'(x) * v(x)

This means you will ALWAYS get the Power Rule derivative for the xⁿ function for any value of n.

Being namely:

( xⁿ )' = n x^n-1

Thus, in general for any value of n:

ƒ'(x) = [ xⁿ sin( x ) ]' = xⁿ cos( x ) + n x^n-1 sin( x )

I hope this helps! Message me in the comments with any questions, comments, or concerns about how I went about this problem!