# How do you use the chain rule to differentiate #y=x^2tan(1/x)#?

You need the chain rule and the product rule here:

Substitute this in the equation above and you have:

We can simplify this expression as:

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To differentiate ( y = x^2 \tan(1/x) ) using the chain rule, follow these steps:

- Identify the outer function and the inner function. In this case, the outer function is ( x^2 ) and the inner function is ( \tan(1/x) ).
- Differentiate the outer function with respect to ( x ). The derivative of ( x^2 ) is ( 2x ).
- Treat the inner function as a separate function ( u = \tan(1/x) ) and differentiate it with respect to ( x ) using the chain rule. The derivative of ( \tan(u) ) is ( \sec^2(u) ) times the derivative of the inner function ( u ) with respect to ( x ).
- Compute the derivative of the inner function ( u = \tan(1/x) ). Let ( u = \tan(1/x) ) and differentiate ( u ) with respect to ( x ).
- Apply the chain rule to find ( \frac{du}{dx} ). Since ( u = \tan(1/x) ), we have ( \frac{du}{dx} = \sec^2(1/x) \cdot \frac{d}{dx}(1/x) ).
- Differentiate ( \frac{1}{x} ) to find ( \frac{d}{dx}(1/x) ). The derivative of ( \frac{1}{x} ) is ( -\frac{1}{x^2} ).
- Substitute the results from steps 2, 5, and 6 into the chain rule formula: ( \frac{dy}{dx} = 2x \cdot \sec^2(1/x) \cdot (-\frac{1}{x^2}) ).
- Simplify the expression to get the final result for ( \frac{dy}{dx} ).

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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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