Partial Derivatives

Partial Derivatives

1. Partial derivatives introduction

   • Do we take the derivative of an expression like this and there's a certain method called a partial derivative
     • You have something like f of X is equal to x squared
       • Derivative DF DX, sketch out a graph so you know this axis represents our output this will be here represents our input and x squared has a certain parabolic.
         • Input x equals 1, DX interpret as just a little nudge in the X direction and it's kind of the size of that nudge and then DF DF is the 6resulting
           • DF DX and interpret that is saying hey how does a tiny change in the input in the X Direction influence
             • How does that influence the output
               • Just like a number line as our output and somehow we're thinking about the function as mapping points on the plane to the number line
               • Changes because it only cares about the X direction neither does this this only cares about the y direction so each one is only a small part of the story
                 • Derivative of sine of Y there's no X's in there so that remains the sine of Y if you're in clear and now this is a more general formula if you plugged in 1

2. Graphical understanding of partial derivatives

   • What the tangent line does and that's, feeling for the slope and you look at that value and the line itself looks like it has a slope of about negative
     • Points on the graph where x equals negative one and now when we take the partial derivative we're going to interpret it as a slice as the slope of the as a slope of this resulting curve
       • I see x squared times y it's considering x squared to be a constant now so it looks at then says X Y or a constant while your the variable constant times the variable the derivative is just equal to that
         • Tlking about the partial derivative as being the slope of the slice of a graph which is great if you're looking at a function that has a two

3. Formal definition of partial derivatives

   • Derivative and how you compute it how
     • Applies to a function that has a multi variable
       • Function like f of x equals x squared
         • Output f of X is what the y-axis represents here and then you're thinking of DF as being the resulting nudge here the resulting change to the function so
           • Your partial f partial x sorry your change in the X direction and this is that resulting change for the function
             • With respect to X and the point B the point B just remains unchanged right so this is this is you evaluating it kind
               • Derivative the formal definition of the partial derivative looks very similar to the formal definition of the derivative but I just I just always think about this as spelling out what we mean by partial Y and partial F and kind of 7:34spelling out why it is that the Leibniz came up with this notation in the first place well I don't know if Leibniz came up with the partials but the the DF DX portion

4. Symmetry of second partial derivatives

   • Second partial derivatives
     • Derivative of F with respect to X and what you do for that X looks like a variable as far as this direction is concerned Y looks like a constant
       • Variable the derivative of which is negative sine times that variable and Y squared here just looks like a constant so it just stays constant at Y squared
         • Y it looks here it says sine of X looks like a variable 2y looks like a constant so what we end up getting is derivative of sine of X cosine X multiplied by that 2y
           • Derivative of F when you do it the other way around when instead of doing X and then Y you do Y and then X partial X that these guys are equal to each other
             • Schwartz's theorem where if the second partial derivatives of your function are continuous at the relevant point that's the circumstance
               • Instead of saying partial squared F partial x squared they'll just write it as partial and then X X and over here this would be partial