https://youtu.be/12OY1b3DYHQ

In this video I show one of the most useful and important concepts in all of math, Implicit Differentiation. This concept is basically applying the derivative to implicit functions, which are functions that have not been explicitly solved with y or f(x) one side of the equation and all of the x or independent variables on the other side. Some equations are very hard to solve for y or f(x) explicitly while some are impossible, but solving there derivatives is much easier because you don't have to solve for y explicitly.

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# Implicit Differentiation – An Introduction

![Implicit Differentiation Resized AI.jpg](https://files.peakd.com/file/peakd-hive/mes/23tSyz4YxKv5vebmbn6AZKgoGqiQ8z2Gcwm6VJxRU8kB2z7nGaw4hKRThkHUAYptSvJv8.jpg)

## Explicit Functions

Functions where the y and x variables are on separate sides of the equal sign are explicit functions, such as the following examples:

>y = f(x) 
>y = x cos(x).

## Implicit Functions

Functions where the x and y variables appear on the side side of the equal sign are implicit functions.

Some can readily be solved as explicit functions but some are difficult or impossible to do so.

![image.png](https://files.peakd.com/file/peakd-hive/mes/23t8CSPFBuN14Y6xXUFELCSbAL1uFKYnmgFrLHk7ar3LVy2SWPFZngTu3h9QpPXbHDMpT.png)

# Implicit Differentiation

To get the derivative y' = dy/dx, we don't need to solve for y "explicitly". 

Instead we can take the derivative, in terms of x, on both sides:

![image.png](https://files.peakd.com/file/peakd-hive/mes/23t8CfKPvnnf17vq6Lc418Tf9wq9NFKvcdkrgepK2tkNXcXU2AUxkJQ9DzDwwNe8cgPb5.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/23t8DBnQFJTeEkhn51TGt3cTZRvpAPUccnfQxG5dKc6FVtUv8CW4Fs2nJ91DBzz7TVxPk.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/23t8DBnQFAcRVWRfAxAE21YgHUASzmvVYgMGZ4YoQpUaR3Vmi6JbgE35pNb1dFHAVsdJS.png)

## Example

Find the derivative of x<sup>3</sup> + y<sup>3</sup> = 6xy

**Solution:**

![image.png](https://files.peakd.com/file/peakd-hive/mes/23tHbKgySTDoGoJagSPZWKVKMRJniYs3D3X9GHZjCtey6SyqWaFcymTs9rmuXktW3Rr5j.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/23t8CfKPvcSgn7HzVFxfWrtEzezAykyANgSccf7efDPY5FnMh66QBhNH5WskH8cYas1Mo.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/Eo4KDWWre3cmXpt6TU4YRV5pGu8JEZvBgfNzuFHeKca2AtcBAfhfW5tTZBgYedhK2WV.png)