How do you differentiate #f(x)=e^tan(1/x^2) # using the chain rule?
Use substitution
Another approach
Now differentiate both sides
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To differentiate ( f(x) = e^{\tan(1/x^2)} ) using the chain rule:
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Find the derivative of the outer function ( e^u ), where ( u = \tan(1/x^2) ): [ \frac{d}{du}(e^u) = e^u ]
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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) ]
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Compute the derivative of ( \frac{1}{x^2} ): [ \frac{d}{dx}\left(\frac{1}{x^2}\right) = -2x^{-3} ]
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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}) ]
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Simplify the expression if needed.
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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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