How do you differentiate #f(x)=e^tan(1/x^2) # using the chain rule?

Answer 1

Use substitution

#f(x) = e^(tan(1/x^2))# Assume #t=1/x^2# then #p = tan(t)# then #f(p) = e^p# #(df)/(dx) = (df)/(dp) * (dp)/(dt) *(dt)/(dx)# # = e^p*sec^2t *(-2/x^3)# Now back substitute
# = e^(tan(1/x^2)) \times sec^2(1/x^2) \times ((-2)/x^3)#

Another approach

Take #ln# on both sides #ln(f(x)) = tan(1/x^2)#

Now differentiate both sides

#1/f \times (df)/(dx) = sec^2(1/x^2) \times ((-2)/x^3)# # (df)/(dx) = f \times sec^2(1/x^2) \times ((-2)/x^3)# # = e^(tan(1/x^2)) \times sec^2(1/x^2) \times ((-2)/x^3)#
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Answer 2

To differentiate ( f(x) = e^{\tan(1/x^2)} ) using the chain rule:

  1. Find the derivative of the outer function ( e^u ), where ( u = \tan(1/x^2) ): [ \frac{d}{du}(e^u) = e^u ]

  2. Find the derivative of the inner function ( \tan(1/x^2) ): [ \frac{d}{dx}(\tan(1/x^2)) = \frac{d}{dx}\left(\tan\left(\frac{1}{x^2}\right)\right) ] Apply the chain rule again: [ = \sec^2\left(\frac{1}{x^2}\right) \cdot \frac{d}{dx}\left(\frac{1}{x^2}\right) ]

  3. Compute the derivative of ( \frac{1}{x^2} ): [ \frac{d}{dx}\left(\frac{1}{x^2}\right) = -2x^{-3} ]

  4. Substitute the results back into the chain rule: [ \frac{d}{dx}(f(x)) = e^{\tan(1/x^2)} \cdot \sec^2\left(\frac{1}{x^2}\right) \cdot (-2x^{-3}) ]

  5. Simplify the expression if needed.

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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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