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Consider the equation
This is not a function, but rather a relation. Its graph looks like this:
But if you drag the point around, you see that the curve still has a well defined slope and tangent! How do we find it?
Since both sides of the equation are equal, they both have to change by the same amount when we make a small change to ​x.
We know what the derivative of ​x2​ is:
And ​dxd​(1)=0​ because ​1​ is a constant.
But what about ​2y2? Well since ​y​ depends on ​x, we use the chain rule:
But ​dyd(y2)​=2y, so
Plugging everything back in gives:
We can rearrange to find
This is an example of a technique called implicit differentiation. It probably seems super weird right now, but after a few practice examples this will start to feel routine.
You might notice that in this case ​dxdy​=yx​, which depends on ​x​ and ​y. This is often the case in implicit differentiation, simply because the curve is not a pure function of ​x, so the slope is usually not either (I say usually because sometimes the curves are symmetric).
Consider the equation
This is not a function, but rather a relation. Its graph looks like this:
But if you drag the point around, you see that the curve still has a well defined slope and tangent! How do we find it?
Since both sides of the equation are equal, they both have to change by the same amount when we make a small change to ​x.
We know what the derivative of ​x2​ is:
And ​dxd​(1)=0​ because ​1​ is a constant.
But what about ​2y2? Well since ​y​ depends on ​x, we use the chain rule:
But ​dyd(y2)​=2y, so
Plugging everything back in gives:
We can rearrange to find
This is an example of a technique called implicit differentiation. It probably seems super weird right now, but after a few practice examples this will start to feel routine.
You might notice that in this case ​dxdy​=yx​, which depends on ​x​ and ​y. This is often the case in implicit differentiation, simply because the curve is not a pure function of ​x, so the slope is usually not either (I say usually because sometimes the curves are symmetric).