# How do you find the derivative of #tan(x/y)=x+y#?

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Use implicit differentiation:

You need the chain rule on the tangent part:

Distribute on the left side:

Get a common denominator in both the numerator and the denominator:

Simplify the complex fraction (I think of it as keep, change, turn):

I factored the negative out of the denominator and distributed it to the numerator:

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To find the derivative of ( \tan\left(\frac{x}{y}\right) = x + y ), we can use implicit differentiation:

[ \frac{d}{dx} \left[ \tan\left(\frac{x}{y}\right) \right] = \frac{d}{dx} (x + y) ]

Using the chain rule for the left side and the sum rule for the right side, we get:

[ \sec^2\left(\frac{x}{y}\right) \cdot \frac{1}{y} = 1 ]

Now, solve for ( \frac{dy}{dx} ):

[ \frac{dy}{dx} = y \sec^2\left(\frac{x}{y}\right) ]

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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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