On Partial Differentiation

This apparently isn’t on my a-level syllabus (or at least integration of this type isn’t) but, thanks to my pure teacher - who is ancient, a genius and a legend, I now know how to do it…

So differentiating with two variables

Lets take a simple equation like

To differentiate this we need to differentiate in respect to x and y separately. When we do this we are effectively ignoring x or y and taking a tangent of a line as if x or y was a constant so you just have a plane.

So if we want the rate of change of z in terms of x then we treat y as a constant.

So as you can see we have differentiated the x term as normal but the y term has been treated like a constant and has thus disappeared. This is our first partial derivative. Our second partial derivative is,

So therefore at x=4, y=2.

The gradient in the y-direction is 4 and in the x-direction is 8.

Now from this we can work out the tangent plane at x=4, y=2.

The formula for this will be,

Where m and n are the gradients in their respective directions.

So when x=4, y=2.

And

So

So our tangent plane is

To verify this we can solve

And

simultaneously and as the original formula is of degree 2 the should only be one solution, at (4,2,20).

~ by Nathan Edwards on April 23, 2008.