1.6 Partial derivatives

Partial derivatives

A Partial derivative is defined as the derivative of the function w.r.t. one of the variables while holding the others constant $$\frac{\partial f}{\partial x}=\underset{\Delta x\to 0}{lim}\frac{f\left(x+\Delta x,t\right)-f\left(x,t\right)}{\Delta x}$$ $$\frac{\partial f}{\partial t}=\underset{\Delta t\to 0}{lim}\frac{f\left(x,t+\Delta t\right)-f\left(x,t\right)}{\Delta t}$$ Some examples: $$f(x,t)=3x^2+x{t}^2$$ $$\frac{\partial f}{\partial x}=6x+t^2$$ $$\frac{\partial f}{\partial t}=2xt$$ $$\frac{{\partial }^{2}f}{\partial x^2}=6$$ $$\frac{{\partial }^{2}f}{\partial t^2}=2x$$ $$\frac{{\partial }^{2}f}{\partial x\partial t}=\frac{{\partial }^{2}f}{\partial t\partial x}=2t$$