prove Gradient(f*g)=(gradient f)*(g) + (gradient g)*(f)

[mlane]

so far I have grad f = fx + fy (fx=partial derivative x, etc.)
grad g= gx + gy
grad (fg)=(f*g)x+(f*g)y?
grad g*(f)= f(gx+gy) and grad f*(g)=g(fx+fy)

Might this be easier if I use df/dx and dg/dx...etc?
this theory seemed to work when I gave f and g a function but without the actual functions I am getting confused.
am i correct in trying to multiply the two.

Is grad (f*g) the same as grad (f) *grad(g)? I think I need to multiply the functions first then figure the gradient.
It doesn't make sense with out an equation?
Could use some pointers in the right direction.


[/tex]

 

[pka]

\(\displaystyle \L \begin{array}{rcl}
\frac{\partial }{{\partial x}}\left( {fg} \right) & = & \frac{{\partial f}}{{\partial x}}g + f\frac{{\partial g}}{{\partial x}}\quad \& \quad \frac{\partial }{{\partial y}}\left( {fg} \right) = \frac{{\partial f}}{{\partial y}}g + f\frac{{\partial g}}{{\partial y}} \\
& = & \frac{{\partial f}}{{\partial x}}g + f\frac{{\partial g}}{{\partial x}} + \frac{{\partial f}}{{\partial y}}g + f\frac{{\partial g}}{{\partial y}} \\
& = & g\left( {\frac{{\partial f}}{{\partial x}} + \frac{{\partial f}}{{\partial y}}} \right) + f\left( {\frac{{\partial g}}{{\partial x}} + \frac{{\partial g}}{{\partial y}}} \right) \\
& = & g\left( {\nabla f} \right) + f\left( {\nabla g} \right) \\
\end{array}.\)

 

[mlane]

gradient proof

Thanks