Let's compute partial derivatives!

Compute the partial derivatives of the following functions:

• $f(x,t) = \sqrt{x} \ln t$
• $z = (2x+3y)^{10}$
• $z = \tan (xy)$
• $u = \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2}$
• $u = \sin( x_1 + 2 x_2 + \cdots + n x_n)$

More derivatives:

Find $\partial z/ \partial x$ and $\partial z/ \partial y$

• $z = f(x) g(y)$
• $z = f(x/y)$
• $z = f(x+y)$
• $z = f(x) + g(x) $
• $z = f(x g(y))$

Laplace's equation: $u_{xx} +u_{yy} = 0$

Determine whether each of the following functions is a solution of Laplace's equation.

• $u = x^2 + y^2$
• $u = x^2 - y^2$
• $u = x^3 + 3xy^2$
• $u = \ln \sqrt{x^2 + y^2}$
• $u = e^{-x} \cos y - e^{-y} \cos x$

Kinetic energy

The kinetic energy of a body with mass $m$ and velocity $v$ is $K = \frac{1}{2} mv^2$. Show that $$\frac{\partial K}{\partial m} \frac{\partial^2 K}{\partial v^2} = K.$$