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What is the derivative of #tanx^2#?

Answer 1

#2xsec^2(x^2)#

We'll have to use the chain rule here. The chain rule, in plain English, says that the derivative of a compound function (like #tan(x^2)#, (which is the #x^2# function inside the #tan(x)# function - that's what makes it compound), is the derivative of the "inside" function multiplied by the derivative of the entire function. In mathspeak, we say the derivative of #f(g(x))=f'(g(x))*g'(x)#.

In our case, the "inside" function is #x^2#, and the derivative of #x^2# is, of course, #2x#. The entire function is #tan(x^2)#, and we know the derivative of #tan(x)# is #sec^2(x)#; so the derivative of #tan(x^2)# is #sec^2(x^2)#. Multiplying these two derivatives together gives #2xsec^2(x^2)#, which is our final result.

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Answer 2

Charles Anctil

The derivative of ( \tan(x^2) ) can be found using the chain rule. The derivative is ( 2x \sec^2(x^2) ).

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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.