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Say you have a function

The derivative of this is then

Now, what is *dx/dy*? The derivative *dy/dx* gives us how fast *y* is changing as we change *x*, and *dx/dy* gives us how fast *x* is changing as we change *y*. We can find this is several ways.

**Using the inverse function**

The most natural way might be to make *x* the subject and then take the derivative. We thus switch the roles of *x* and *y*. We get

and then

Here we have to remember that *x* is now a function of our variable *y*.

**Using the reciprocal**

If *dy/dx*=*m*, could it be that *dx/dy=*1*/m*? Could it be that simple?

Normally we see dy/dx as one symbol the derivative of y with respect to *x*. But could the parts *dy* and *dx* take on roles as separate entities? Let us say we have some function, and that the derivative at a particular point is *m*, and if we draw a tangent line at that point, then the tangent line would have the slope *m* at every point.

We may then choose to let *dy* and *dx* represent any two values such that *dy/dx*=*m*. If so then we may start to manipulate them as any variables and thus do the following

We can also show this by

The last step can be done since, if the derivative exists, both Δ*x* and Δ*y* will go toward 0 simultaneously.

This would now give us, for our function, *y*=*x*^{2} that

but since

we get

**Using implicit differentiation**

We may also take the derivative of *y*=*x*^{2} with respect to *x* to get

or

**Another example**

Let

- The derivative of the inverse function

So

- The reciprocal

The derivative *dy/dx* is *y*´=3*x*^{2}, so

But

so

and thus

- The implicit differential

The derivative with respect to y of

Gives us

So

And then we do as above.

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