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Math Help - Directional derivative definition via difference quotients

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    Directional derivative definition via difference quotients

    Directional derivative, definition via difference quotients -
    I thought to demostrate in this way :

     w={f}(x_o,y_o)

     \frac{dw}{ds}=\frac{d{f}(x_o,y_o)}{ds} = \lim_{h_x \rightarrow 0, h_y \rightarrow 0}{\frac{f(x_o + h_x, y_o+h_y) - f(x_o,y_o)}{\sqrt{h_x^2+h_y^2}}}

    or

     \frac{d{f}(x_o,y_o)}{ds} = \lim_{h \rightarrow 0}{\frac{f(x_o + h*cos(\phi), y_o+h*sin(\phi))) - f(x_o,y_o)}{h}}

    -----------------

    if It increase before  \Delta x = h_x , from P_o \rightarrow P_x then  \Delta w_a=f{(x_o+h_x,y_o)}-f{(x_o,y_o)}

    and after increase  \Delta y = h_y from P_x \rightarrow P_1 then  \Delta w_b=f{(x_1,y_o+h_y)}-f{(x_1,y_o)}

    in every case this is equal to P_o \rightarrow P_1 that is  \Delta w=\Delta w_a + \Delta w_b

    via difference quotients it can write :

     \frac{f(x_o + h_x, y_o+h_y) - f(x_o,y_o)}{h}  = \frac{f(x_o + h_x, y_o) - f(x_o,y_o)}{h} + \frac{f(x_1, y_o+h_y) - f(x_1,y_o)}{h}

    but  h_x=h*cos(\phi) and  h_y=h*sin(\phi)

    then

     \frac{f(x_o + h_x, y_o+h_y) - f(x_o,y_o)}{h}  = \frac{f(x_o + h_x, y_o) - f(x_o,y_o)}{\frac{h_x}{cos(\phi)}} + \frac{f(x_1, y_o+h_y) - f(x_1,y_o)}{\frac{h_y}{sin(\phi)}}

     \frac{f(x_o + h_x, y_o+h_y) - f(x_o,y_o)}{h}  = \frac{f(x_o + h_x, y_o) - f(x_o,y_o)}{h_x}*cos(\phi) + \frac{f(x_1, y_o+h_y) - f(x_1,y_o)}{h_y}*sin(\phi)

     \frac{\Delta w}{h} = \frac{f(x_o + h_x, y_o) - f(x_o,y_o)}{h_x}*cos(\phi) + \frac{f(x_1, y_o+h_y) - f(x_1,y_o)}{h_y}*sin(\phi)

     \frac{dw}{ds}  = \lim_{h_x \rightarrow 0}{\frac{f(x_o + h_x, y_o) - f(x_o,y_o)}{h_x}*cos(\phi)} + \lim_{h_y \rightarrow 0}{\frac{f(x_1, y_o+h_y) - f(x_1,y_o)}{h_y}*sin(\phi)}

    the first limits when  h_x \rightarrow 0 means x_1 \rightarrow x_o
    then the second difference quotient can be write :

     \frac{f(x_o,y_o)}{ds} = f_x(x_o,y_o)*cos(\phi) + \lim_{h_y \rightarrow 0}{\frac{f(x_o, y_o+h_y) - f(x_o,y_o)}{h_y}*sin(\phi)}

    and all thogether :

     \frac{df(x_o,y_o)}{ds} = f_x(x_o,y_o)*cos(\phi) + f_y(x_o,y_o)*sin(\phi)

    this is a wrong way ?
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