f-R-R-g-R-R-d-fg-dx-df-dx-dg-dx-d-f-2-dx-df-dx-df-dx-d-g-2-dx-

Question Number 767 by 123456 last updated on 09/Mar/15

f : R \to R

g : R \to R

\frac{d (f g)}{d x} = \frac{d f}{d x} \cdot \frac{d g}{d x}

\frac{d (f^{2})}{d x} = \frac{d f}{d x} \cdot \frac{d f}{d x}

\frac{d (g^{2})}{d x} = ?

Commented by 123456 last updated on 09/Mar/15

\frac{d (f^{2} g^{2})}{d x} = \frac{d (f^{2})}{d x} g^{2} + f^{2} \frac{d (g^{2})}{d x} = {(\frac{d f}{d x} g)}^{2} + f^{2} \frac{d (g^{2})}{d x}

\frac{d (f^{2} g^{2})}{d x} = 2 \frac{d (f g)}{d x} f g = 2 \frac{d f}{d x} \cdot \frac{d g}{d x} f g

{(\frac{d f}{d x} g)}^{2} + f^{2} \frac{d (g^{2})}{d x} = 2 \frac{d f}{d x} \cdot \frac{d g}{d x} f g

f^{2} \frac{d (g^{2})}{d x} = 2 \frac{d f}{d x} \cdot \frac{d g}{d x} f g - {(\frac{d f}{d x} g)}^{2}

\frac{d (g^{2})}{d x} = \frac{1}{f^{2}} [2 \frac{d f}{d x} \cdot \frac{d g}{d x} f g - {(\frac{d f}{d x} g)}^{2}]

= \frac{1}{f^{2}} \cdot \frac{d f}{d x} g (2 \frac{d g}{d x} f + \frac{d f}{d x} g)

= \frac{1}{f^{2}} \cdot \frac{d f}{d x} g (f \frac{d g}{d x} + f \frac{d g}{d x} + \frac{d f}{d x} g)

= \frac{1}{f^{2}} \cdot \frac{d f}{d x} g [f \frac{d g}{d x} + \frac{d (f g)}{d x}]

= \frac{1}{f^{2}} \cdot \frac{d f}{d x} g (f \frac{d g}{d x} + \frac{d f}{d x} \cdot \frac{d g}{d x})

= \frac{1}{f^{2}} \cdot \frac{d f}{d x} \cdot \frac{d g}{d x} g (f + \frac{d f}{d x})

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