# How do you differentiate #e^y cos(x) = 8 + sin(xy)#?

Start by making the assumption that this equation "implicitly defines"

(If you prefer, you could write this as:

This equation, complicated as it looks, can be solved for

Now get the terms involving

Now divide both sides by

(alternatively,

But what does this mean? The original equation

I graphed this equation using ContourPlot on Mathematica (Wolfram Alpha gave me trouble with it) and got the following picture, which seems to pass the vertical line test when

The derivative we found,

What's an example of a known point? The simplest one I could think of is

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To differentiate the equation e^y cos(x) = 8 + sin(xy) with respect to x, you would use the product rule and chain rule. First, differentiate each term separately. The derivative of e^y with respect to x is e^y(dy/dx), and the derivative of cos(x) with respect to x is -sin(x). Then, using the product rule, differentiate 8 with respect to x (which is 0) and sin(xy) with respect to x, which becomes y(cos(xy))(dy/dx) + xcos(xy). Combine all the derivatives to get the final result.

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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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