Partial Derivatives

The partial derivative of $f(x,y)$ with respect to $x$ at the point $(a,b)$ is $f_x(a,b)={\lim }_{h\to 0}\frac{f(a+h,b)-f(a,b)}{h}$ if this limit exists.

Tip

In practice, we simply treat the other variables as constants when we take the partial derivative.

Example: If $f(x,y)=x\cos y+2x^{3}y$, then $\frac{\partial f}{\partial x}=f_x=\cos y+6y{x}^2$ $\frac{\partial f}{\partial y}=f_y=-x\sin y+2x^3$

Clairaut's Theorem

If $f_x,f_y$ and $f_{xy}$ exist near (a,b), and if $f_{xy}$ is continuous at (a,b), then $f_{yx}(a,b)$ also exists, and in fact $f_{yx}(a,b)=f_{xy}(a,b)$.

Related

• Directional Derivatives
• Higher-Order Derivatives