What is the second, third, fourth and fifth derivative of tan(x)?

Answer 1

I'm not sure if this is the quicker or easier route, but it is a viable one:

#y=tanx#
#y^((1))=1+tan^2x=1+y^2#
#y^((2))=2y*y^((1))=2y*(1+y^2)=2y+2y^3#
#y^((3))=2y^((1))+6y^2*y^((1))=2(1+y^2)+6y^2(1+y^2)=#
#=2+2y^2+6y^2+6y^4=2+8y^2+6y^4#
#y^((4))=16yy^((1))+24y^3y^((1))=16y(1+y^2)+24y^3(1+y^2)=#
#=16y+16y^3+24y^3+24y^5=16y+40y^3+24y^5#
#y^((5))=16y^((1))+120y^2y^((1))+120y^4y^((1))=#
#=16(1+y^2)+120y^2(1+y^2)+120y^4(1+y^2)=#
#=16+16y^2+120y^2+120y^4+120y^4+120y^6=#
#=16+136y^2+240y^4+120y^6=#
#=16+136tan^2x+240tan^4x+120tan^6x#.

I'm hoping it's accurate and enjoyable.

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Answer 2

The second derivative of ( \tan(x) ) is (\sec^2(x)).

The third derivative of ( \tan(x) ) is (2\sec^2(x)\tan(x)).

The fourth derivative of ( \tan(x) ) is (2\sec^2(x)(2\tan^2(x)+1)).

The fifth derivative of ( \tan(x) ) is (2\sec^2(x)(6\tan^3(x)+3\tan(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.

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.

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.

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.

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