# How do you differentiate #f(x)=tan(1-3x) # using the chain rule?

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# - 3 sec^2(1 - 3x ) #

differentiating using the 'chain rule ' :

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To differentiate ( f(x) = \tan(1 - 3x) ) using the chain rule, follow these steps:

- Identify the outer function (( \tan(x) )) and the inner function (( 1 - 3x )).
- Differentiate the outer function with respect to the inner function.
- Differentiate the inner function with respect to ( x ).
- Multiply the results from steps 2 and 3 to obtain the derivative.

The derivative of ( f(x) ) is:

[ f'(x) = \frac{d}{dx}[\tan(1 - 3x)] = \frac{d}{d(1 - 3x)}[\tan(x)] \cdot \frac{d}{dx}(1 - 3x) ]

Now, differentiate each part:

[ \frac{d}{d(1 - 3x)}[\tan(x)] = \sec^2(1 - 3x) ] [ \frac{d}{dx}(1 - 3x) = -3 ]

Multiply these results:

[ f'(x) = -3 \sec^2(1 - 3x) ]

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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.

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