Derivatives in Several Variables
When a function has several inputs, you differentiate by freezing all but one variable and asking how fast the output moves.
One variable at a time
The partial derivative with respect to x treats y as a constant and differentiates normally, using a limit with a small h added to x alone. For the function x squared times y, the x derivative is 2xy and the y derivative is x squared. Each partial measures the slope of the surface along one axis direction.
\frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x+h,\, y) - f(x,\, y)}{h}Reading the rates
Evaluate the example function x squared y at the point (3, 2): the x derivative 2xy gives 12, and the y derivative x squared gives 9. So nudging x up by a tiny amount raises the output about 12 times the nudge, while the same nudge in y raises it about 9 times that amount. The two slopes are generally different, which is why the surface tilts differently along each axis.
Tangent plane
Both partials together build the tangent plane, the flat sheet that best hugs the surface near a point. Its formula is the value at the point plus the x slope times the x offset plus the y slope times the y offset. Mixed partials like fxy also commute when they are continuous, so order of differentiation does not matter in that case.
L(x,y) = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)Flashcard
Rate of change of f in the x direction
Check your understanding
If f(x, y) = x cubed times y squared, what is the partial derivative of f with respect to x?