Master Partial Differentiation

To find the rate of change of the volume with respect to radius ($r$), we differentiate the volume equation with respect to $r$, keeping the height ($h$) constant. The first partial derivative of $V$ with respect to $r$ is $\frac{\partial V}{\partial r} = 2\pi rh$.

To find the rate of change of the volume with respect to height ($h$), we differentiate the volume equation with respect to $h$, keeping the radius ($r$) constant. The first partial derivative of $V$ with respect to $h$ is $\frac{\partial V}{\partial h} = \pi r^2$.

The formula for the volume of a cylinder is $V = \pi r^2h$.

Some applications of partial derivatives include optimization problems, finding tangent planes, determining critical points, and analyzing rates of change in multivariable functions.

Euler's Theorem for Homogeneous functions states that if a function $f(x_1, x_2, ..., x_n)$ is homogeneous of degree $k$, then $x_1\frac{\partial f}{\partial x_1} + x_2\frac{\partial f}{\partial x_2} + ... + x_n\frac{\partial f}{\partial x_n} = kf(x_1, x_2, ..., x_n)$.