A partial derivative changes one coordinate while fixing the others. For f=x²+3xy, f_x=2x+3y and f_y=3x. The gradient collects these derivatives; if f is differentiable, the directional derivative along a unit vector v is grad f·v. Normalise the direction before taking this dot product. A direction vector of length two would double the answer if used without normalisation.
Differentiability means a valid linear approximation, not merely the existence of some partial derivatives at one point. Continuous first partials in a neighbourhood are a sufficient condition. For a composition f(x(t),y(t)), the chain rule gives f_x x′+f_y y′. Second derivatives can introduce both direct and mixed terms; a zero mixed partial alone places no restriction on the pure second partials.
A multiple integral sums contributions over a specified region. Describe the region before choosing iterated limits; nonrectangular bounds may change when the order changes. In polar coordinates area is r dr dtheta, not just dr dtheta. A general coordinate change uses the absolute Jacobian determinant. Sign belongs to oriented vector quantities, while area and volume scaling use a nonnegative factor.
If both first partials of a globally defined function on R² are constant, f_x=a and f_y=b, integrating successively gives f=ax+by+c, a plane. This cannot be inferred from parallel straight level sets alone: e^x has vertical parallel level lines but a curved graph. Nor do f_xy=f_yx=0 force a plane: x²+y² is a counterexample. Distinguish first-derivative constancy from absent mixed dependence and from the shape of selected level sets.
