# Partial Derivatives How can we extend our idea of the derivative to 3 dimensions? #### Partial with respect to $x$ We can envision slicing our surface with a plane (by fixing $y = y_0$), then we can look at the tangent line (red) along our curve (blue):   #### Partial with respect to $y$ We can do the same thing by holding $x$ fixed.   #### General Intuition If we wish to understand how a multivariate function, $f$, changes with respect to a certain variable, we can simply hold all other variables **fixed** and see how $f$ changes with respect to our variable of interest, just as we did in the 1-d case. #### Continuity vs. Partial Derivatives vs. Differentiability Recall that in single variable calculus we required continuity if we wanted differentiability:  Now let us look at the following multivariable function:  We can ask two questions: 1. Is it continuous? 2. Does it have derivatives? We can start by trying to determine if it is continuous. We can start by approaching the point $(0,0)$ along the path $y=0$:  And now we can approach the same point along the path $y=x$:  And we see that we have two different paths, and the limit along those two different paths gives two different values, so the limit does not exist. Because the limit does not exist the function is not continuous. Now, remember in calculus I if we knew the function was not continuous then we would know that the function was not differentiable. However, will that be the same case in multivariable calculus, particularly in this example? First let's look at the partial derivative of $f$ with respect to $x$:  Very interesting! So our partial derivative here *exists*! It is equal to a specific number! The same argument applies for the partial derivative with respect to $y$. So, our take away is that we have an example of a function that: * Is **discontinuous** as $(0,0)$ * Has **partial derivatives** that exist at $(0,0)$ --- References: * [What are derivatives in 3D? Intro to Partial Derivatives](https://www.youtube.com/watch?v=EoEV5-_mLeM&list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd&index=14) * [Continuity vs Partial Derivatives vs Differentiability](https://www.youtube.com/watch?v=6Wi1kT9kR1M&list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd&index=15)