How do you differentiate #f(x)=xtan3x+x^3tanx# using the product rule?

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Answer 1 · 2015-12-15T20:55:58

Find both derivatives separately, each using the product rule.

#d/dx[xtan3x]=tan3xd/dx[x]+xd/dx[tan3x]#

#d/dx[x]=1#

#d/dx[tan3x]=sec^2 3x*d/dx[3x]=3sec^2 3x#

Thus,

#d/dx[xtan3x]=tan3x+3xsec^2 3x#

——————————

#d/dx[x^3tanx]=tanxd/dx[x^3]+x^3d/dx[tanx]#

#d/dx[x^3]=3x^2#

#d/dx[tanx]=sec^2x#

Thus,

#d/dx[x^3tanx]=3x^2tanx+x^3sec^2x#

——————————

Add the two to find that

#f'(x)=tan3x+3xsec^2 3x+3x^2tanx+x^3sec^2x#