BobHans-find-the-formula-d-3-dx-3-g-f-x-

Question Number 107647 by bobhans last updated on 12/Aug/20

≍ B ob H ans ≍

find the formula \frac{d^{3}}{{dx}^{3}} [g (f (x))]

Answered by john santu last updated on 12/Aug/20

JS

(1) \frac{d}{d x} g (f (x)) = \frac{d}{d u} g (u) \frac{d}{d x} f (x) = g^{(1)} f^{(1)}

= g^{(1)} (f (x)) f^{(1)} (x)

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

= \frac{d}{d x} [\frac{d}{d u} g (u) \frac{d}{d x} f (x)]

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

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

= g^{(1)} f^{(2)} + g^{(2)} {[f^{(1)}]}^{2}

(3) \frac{d^{3}}{{d x}^{3}} g (f (x)) = g^{(1)} f^{(3)} + 3 g^{(2)} f^{(1)} f^{(2)} + g^{(3)} {[f^{(1)}]}^{3} .