Partial Derivatives

For functions about more than one variables, we first introduce their partial derivatives instead of their limits, since partial derivative is a simpler concept. We will explain why in the next section.

We take functions about two variables ${\displaystyle f(x,y)}$ here for example.

Rules for Finding Partial Derivatives

We have:

${\displaystyle f_x(x_0,y_0)=\underset{\mathrm{\Delta }x\to 0}{lim}\frac{f(x_0+\mathrm{\Delta }x,y_0)-f(x_0,y_0)}{\mathrm{\Delta }x}}$

That is, to find the partial derivative of ${\displaystyle x}$, we just need to consider ${\displaystyle y}$ as a constant instead of a variable and fibnd the derivative about ${\displaystyle x}$.

e.g.

Find ${\displaystyle f_x(1,2)}$ where ${\displaystyle f(x,y)=x^{2}y+3y^3}$.

We have ${\displaystyle f_x(x,y)=2xy}$,
So ${\displaystyle f_x(1,2)=2\times 1\times 2=4}$

Implications of the Partial Derivatives

The function about two variables is usually a surface in three-space, and the partial derivative of it is actually the tangent line of the surface in ${\displaystyle x}$ or ${\displaystyle y}$ direction.