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How do you differentiate #f(x)=3x+xtanx#?

Answer 1

#f'(x)=3+xsec^2(x)+tan(x)#

#d/dx(f(x))=d/dx(3x+xtan(x))# #f'(x)=d/dx(3x)+d/dx(xtan(x))# #f'(x)=3+x d/dx(tan(x))+d/dx(x)tan(x)# #f'(x)=3+xsec^2(x)+tan(x)#

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Answer 2

Isabella Ackerman

To differentiate ( f(x) = 3x + x\tan(x) ), apply the product rule and the derivative of the tangent function:

Differentiate ( 3x ) with respect to ( x ) to get ( 3 ).
For the term ( x\tan(x) ), differentiate ( x ) to get ( 1 ), then use the product rule for ( u(x) = x ) and ( v(x) = \tan(x) ):
- ( u'(x) = 1 ) (derivative of ( x )).
- ( v'(x) = \sec^2(x) ) (derivative of ( \tan(x) )).
- Apply the product rule: ( (uv)' = u'v + uv' ).
- ( (x\tan(x))' = (1)\tan(x) + x(\sec^2(x)) ).
- Simplify to get ( \tan(x) + x\sec^2(x) ).

Combining these results, the derivative of ( f(x) ) is ( f'(x) = 3 + \tan(x) + x\sec^2(x) ).

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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.