How do you differentiate # f(x) = tan^2(3/x) #?

Answer 1

#f'(x)=(-6sec^2(3/x)tan(3/x))/x^2#

The first issue is the squared function. This will require the chain rule. Treat the problem like you would if it were #x^2#, which gives a derivative of #2x#, except that this must be multiplied by the derivative of the function that was squared as well.
#f'(x)=2tan(3/x)*d/dx[tan(3/x)]#
To differentiate the tangent function, use the chain rule again. Recall that the derivative of #tanx# is #sec^2x#.
#f'(x)=2tan(3/x)sec^2(3/x)*d/dx[3/x]#
To differentiate #3/x#, rewrite it as #3x^-1# and then use the power rule (no chain rule needed here).
#f'(x)=2tan(3/x)sec^2(3/x)(-3x^-2)#

This could be rephrased as

#f'(x)=(-6sec^2(3/x)tan(3/x))/x^2#
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Answer 2

To differentiate ( f(x) = \tan^2\left(\frac{3}{x}\right) ), you would use the chain rule. The derivative of (\tan(u)) with respect to (u) is ( \sec^2(u) ), and the derivative of ( \frac{3}{x} ) with respect to (x) is ( -\frac{3}{x^2} ). Applying the chain rule, we get:

[ \frac{d}{dx}\left[\tan^2\left(\frac{3}{x}\right)\right] = 2\tan\left(\frac{3}{x}\right)\sec^2\left(\frac{3}{x}\right)\left(-\frac{3}{x^2}\right) ]

Simplifying further:

[ = -\frac{6\tan\left(\frac{3}{x}\right)}{x^2}\sec^2\left(\frac{3}{x}\right) ]

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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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