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Lets say we have the relationship y is equal to cosine of five x minus three y

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and what I want to find is the rate of which y is changing with respect to x

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and we'll assume that y is a function of x

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So lets do what weve always been doing

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lets apply the derivative operator to both sides of this equation

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On the left side of the equation, right here, we get dy/dx is equal to

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Now here on the right hand side were going to apply the chain rule

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The derivative of the cosine of something with respect to that something is going to be equal to negative sine of that something

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so negative sine of five x minus three y

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and then we have to multiply that by the derivative of that something with respect to x.

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So whats the derivative of that something with respect to x?

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Well the derivative of five x with respect to x is five

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and the derivative of negative three y with respect to x is negative three times dy/dx

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The negative three times the derivative of y with respect to x

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and now we just need to solve for dy/dx

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As you can see with some of these implicit differentiation problems, this is the hard part

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Actually let me make that dy/dx the same color, so we can keep track of it easier

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So this is going to be dy/dx and then I can close the parentheses

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So how can we do it? Its just going to be a little bit of algebra to work through

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Well we can distribute the sine of five x minus 3 y, so let me rewrite everything

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You can distribute the sine of five x minus three y - you get, let me rewrite everything. You're going to get dy/dx. Your distribute the negative sine, youre going to have five x minus three y

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all of this, so youre going to have negative five times the sine of five

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x minus three y and then youre going have the negative times the negative

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You end up with a positive and so youre going to end up with plus three

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times the sine of five x minus three y

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dy/dx

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Now what we can do is subtract three sine of five x minus three y from both sides

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So, this is essentially one dy/dx

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So if we subtract this from both sides we are left with

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So one the left side were going to have a one dy/dx

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and were going to subtract from that three sine of five x minus three y dy/dx's

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Youre going to have one minus three

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Ill keep the color for three for fun

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three sine of five x minus three y dy/dx's

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on the left hand side

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is going to be equal to

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well we subtracted thsi from both sides, so on the right hand side this is going to go away

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So were just going to be left with a negative five sine of five x minus three y

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and were in the home stretch now! To solve for dy/dx

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You just have to divide both sides of the equation by this and we are left with

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dy/dx is equal to

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this thing, negative five times the sine

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of five x minus three y

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all of that over one minus three sine

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of five x minus three y and we are done