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Math Help - Proving the formula for the directional derivatives of the of the sum and dot product

  1. #1
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    Proving the formula for the directional derivatives of the of the sum and dot product

    Define the directional derivative of a function \textbf{f} at \textbf{c} in the direction \textbf{u} by
    \textbf{f}\hspace{0.04in}'(\textbf{c};\textbf{u}) = \lim_{h \rightarrow 0} \frac{\textbf{f}(\textbf{c}+h\textbf{u}) - f(\textbf{c})}{h},
    whenever the limit on the right exists.

    Let \textbf{f} and \textbf{g} be functions with values in \mathbb{\textbf{R}}^m such that the directional derivatives \textbf{f}\hspace{0.04in}'(\textbf{c};\textbf{u}) and \textbf{g}'(\textbf{c};\textbf{u}) exist.

    Prove that the sum \textbf{f+g} and dot product \textbf{f} \bullet \textbf{g} have directional derivatives given by
    (\textbf{f+g})'(\textbf{c};\textbf{u}) = \textbf{f}\hspace{0.04in}'(\textbf{c};\textbf{u}) + \textbf{g}'(\textbf{c};\textbf{u})
    and
    (\textbf{f} \bullet \textbf{g})'(\textbf{c};\textbf{u)} = \textbf{f}(\textbf{c}) \bullet \textbf{g}'(\textbf{c};\textbf{u}) + \textbf{g}(\textbf{c}) \bullet \textbf{f}\hspace{0.04in}'(\textbf{c};\textbf{u}).


    \underline{\textbf{What I've Tried}}
    I've figured out how to prove the sum part already but I'm having trouble with the dot product part. I tried to do it backwards

    f(c) \bullet g'(c;u) + g(c) \bullet f'(c;u)

    = f(c) \bullet \lim_{h \rightarrow 0} \frac{g(c+hu)-g(c)}{h} + g(c) \bullet \lim_{h \rightarrow 0} \frac{f(c+hu)-f(c)}{h}

    = \lim_{h \rightarrow 0} f(c) \bullet \frac{g(c+hu)-g(c)}{h} + \lim_{h \rightarrow 0} g(c) \bullet \frac{f(c+hu)-f(c)}{h}

    =\lim_{h \rightarrow 0} \frac{f(c) \bullet g(c+hu) - f(c) \bullet g(c) + g(c) \bullet f(c+hu) - g(c)\bullet f(c)}{h}

    \textbf{But I'm not sure how that will arrive to}:

    = \lim_{h \rightarrow 0}  \frac{f(c+hu)\bullet g(c+hu) - f(c) \bullet g(c)}{h}

    = \lim_{h \rightarrow 0}  \frac{(f \bullet g)(c+hu) - (f \bullet g)(c)}{h}

    = (f \bullet g)'(c;u)
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  2. #2
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    Re: Proving the formula for the directional derivatives of the of the sum and dot pro

    Hint: Look at proving the product rule for normal, single variable derivatives. There's a term you add and subtract in the numerator to make everything cancel out.
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