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What is the second derivative of #f(x)=tan(3x)#?

Answer 1

Ezra Adams

#f''(x)=18sec^2(3x)tan(3x)#

We will use the chain rule, together with the derivatives:

#d/dx tan(x) = sec^2(x)#

#d/dx 3x = 3#

#d/dx x^2 = 2x#

# d/dx sec(x) = sec(x)tan(x)#

First Derivative: #f'(x) = d/dx tan(3x)#

#= sec^2(3x)(d/dx3x)#

#= 3sec^2(3x)#

Second Derivative:

#f''(x) = d/dx(f'(x))#

#= d/dx(3sec^2(3x))#

#= 3d/dx(sec^2(3x))#

#= 3(2sec(3x))(d/dxsec(3x))#

#=6sec(3x)(sec(3x)tan(3x))(d/dx3x)#

#=18sec^2(3x)tan(3x)#

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Answer 2

Emma Aldridge

The second derivative of ( f(x) = \tan(3x) ) is ( f''(x) = 18\sec^2(3x) ).

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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.