Derivatives

1. The partial derivative is like a derivative while holding other variables constant.

2. Linearization is approximating a function \(f \) at \((x_0,y_0,z_0)\)with a plane, and the equation of the plane is$$z-z_0=\frac{\partial f}{\partial x}(x-x_0)+\frac{\partial f}{\partial y}(y-y_0)$$

3. The function being differentiable at a point means the function can be approximated by a plane at that point.

4. If the partial derivatives are continuous at that point, then the function is differentiable.

5. The differential is $$dz=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy$$

6. The multivariable chain rule is

$$\begin{cases}
\frac{\partial z}{\partial s}=\frac{\partial z}{\partial x}\frac{\partial x}{\partial s}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial s}\\
\frac{\partial z}{\partial t}=\frac{\partial z}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial t}
\end{cases}$$

7.

If \(F(x,y,z)=0\),

$$\frac{\partial z}{\partial x}=-\frac{\partial F}{\partial x}/\frac{\partial F}{\partial z}$$

8. The gradient is a vector that points to the direction with the maximum directional derivative and its magnitude is the the directional derivative in this direction.

$$\nabla f=(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y})$$

9. The gradient is always perpendicular to contour lines/planes(sets of points with the same function value)

10. The tangent plane of \(F(x,y,z)=k\) is

$$ \frac{\partial F}{\partial x}(x-x_0)+\frac{\partial F}{\partial y}(y-y_0)+\frac{\partial F}{\partial z}(z-z_0)=0$$

$$or$$

$$\nabla F\cdot \vec r=0$$

11. The critical point of a function is where it might have a maximum or minimum. The critical point is where

$$\nabla f=\vec 0$$

12. To determine whether a critical point is a minimal of maximum, we calculate

$$D=\frac{\partial^2 f}{\partial x^2}\frac{\partial^2 f}{\partial y^2}-(\frac{\partial^2 f}{\partial x\partial y})^2$$

$$\begin{cases}
D>0\ \&\ \frac{\partial^2 f}{\partial x^2}>0\to min\\
D>0\ \&\ \frac{\partial^2 f}{\partial x^2}<0\to max\\
D=0\to indeterminant\\
D<0\to saddle\\
\end{cases}$$

13. Lagrange Multiplier. If we want to find extrema of \(f(x,y)\) with constraint \(g(x,y)=k\), then we can solve equations

$$\nabla f=\lambda\nabla g$$

$$or$$

$$\begin{cases}
\frac{\partial f}{\partial x}=\lambda \frac{\partial g}{\partial x}\\
\frac{\partial f}{\partial y}=\lambda \frac{\partial g}{\partial y}\\
g(x,y)=k
\end{cases}$$

For multiple constraints

$$\nabla f=\lambda\nabla g+\mu \nabla h$$

14. Clairaut’s theorem, if the function has a continuous second partial derivatives, the order of taking partial derivatives does not affect the result

$$f_{xy}=f_{yx}$$