Partial Derivatives: Computation, Higher Derivatives, and Applications

All flashcards

What is a partial derivative?

It is the derivative of a function of several variables with respect to one selected variable, holding all other variables constant.

What does $f_x$ mean?

$f_x$ denotes the partial derivative of $f$ with respect to $x$, describing its rate of change in the $x$ direction while other inputs are fixed.

What does the limit definition vary?

It changes one coordinate by a small amount $h$ and compares the resulting function value with the original, then takes the limit as $h$ approaches zero.

How is a coordinate partial derivative a directional derivative?

It is the directional derivative in the direction of the corresponding standard unit vector, such as $\mathbf{e}_i$.

What does the $C^1$ condition guarantee locally?

If the first partial derivatives exist and are continuous in a neighborhood, the function is totally differentiable there and its total derivative is continuous.

Quiz questions

1. Under what condition does Clairaut's theorem guarantee that $f_{xy}=f_{yx}$ near a point?Answer: The relevant second partial derivatives are continuous near the point.
2. What quantities are arranged in the Hessian matrix of a function of two variables?Answer: The function's second partial derivatives.
3. If $x=r\cos\theta$ and $y=r\sin\theta$, what is $\frac{\partial f}{\partial r}$ when $\theta$ is fixed?Answer: $f_x\cos\theta+f_y\sin\theta$
4. For $f(x,y)=x^2y+3y^2$, what is $f_{xy}$?Answer: $2x$
5. For $V(r,h)=\frac{\pi r^2h}{3}$ with $h$ depending on $r$, which expression gives $\frac{dV}{dr}$?Answer: $\frac{2\pi rh}{3}+\frac{\pi r^2}{3}\frac{dh}{dr}$