d/(dt)(sqrt(({f)(t))^2+({g}(t))^2+({f}(t)^'(t))^2})

解

\frac{d}{d t} ((f (t))^{2} + (g (t))^{2} + (f (t) t^{^{'}} (t))^{2})

\frac{f ( t ) \frac{d}{d t} ( f ( t ) ) + g ( t ) \frac{d}{d t} ( g ( t ) ) + ( ' + 1 ) t ^{2' + 1} f ( t ) ^{2} + t ^{2 (' + 1)} f ( t ) \frac{d}{d t} ( f ( t ) )}{f ( t ) ^{2} + g ( t ) ^{2} + t ^{2 (' + 1)} f ( t ) ^{2}}

解答ステップ

\frac{d}{d t} ((f (t))^{2} + (g (t))^{2} + (f (t) t^{^{'}} (t))^{2})

連鎖法則を適用:

\frac{1}{2 f ( t ) ^{2} + g ( t ) ^{2} + ( f ( t ) t ^{^{'}} t ) ^{2}} \frac{d}{d t} (f (t)^{2} + g (t)^{2} + (f (t) t^{^{'}} t)^{2})

= \frac{1}{2 f ( t ) ^{2} + g ( t ) ^{2} + ( f ( t ) t ^{^{'}} t ) ^{2}} \frac{d}{d t} (f (t)^{2} + g (t)^{2} + (f (t) t^{^{'}} t)^{2})

\frac{d}{d t} (f (t)^{2} + g (t)^{2} + (f (t) t^{^{'}} t)^{2}) = 2 f (t) \frac{d}{d t} (f (t)) + 2 g (t) \frac{d}{d t} (g (t)) + 2 (^{'} + 1) t^{2^{'} + 1} f (t)^{2} + 2 f (t) \frac{d}{d t} (f (t)) t^{2 (^{'} + 1)}

= \frac{1}{2 f ( t ) ^{2} + g ( t ) ^{2} + ( f ( t ) t ^{^{^{'}}} t ) ^{2}} (2 f (t) \frac{d}{d t} (f (t)) + 2 g (t) \frac{d}{d t} (g (t)) + 2 (^{^{'}} + 1) t^{2^{^{'}} + 1} f (t)^{2} + 2 f (t) \frac{d}{d t} (f (t)) t^{2 (^{^{'}} + 1)})

= \frac{1}{2 f ( t ) ^{2} + g ( t ) ^{2} + ( f ( t ) t ^{^{^{'}}} t ) ^{2}} (2 f (t) \frac{d}{d t} (f (t)) + 2 g (t) \frac{d}{d t} (g (t)) + 2 (^{^{'}} + 1) t^{2^{^{'}} + 1} f (t)^{2} + 2 f (t) \frac{d}{d t} (f (t)) t^{2 (^{^{'}} + 1)})

簡素化

\frac{1}{2 f ( t ) ^{2} + g ( t ) ^{2} + ( f ( t ) t ^{^{'}} t ) ^{2}} (2 f (t) \frac{d}{d t} (f (t)) + 2 g (t) \frac{d}{d t} (g (t)) + 2 (^{'} + 1) t^{2^{'} + 1} f (t)^{2} + 2 f (t) \frac{d}{d t} (f (t)) t^{2 (^{'} + 1)}) : \frac{f ( t ) \frac{d}{d t} ( f ( t ) ) + g ( t ) \frac{d}{d t} ( g ( t ) ) + ( ^{'} + 1 ) t ^{2^{'} + 1} f ( t ) ^{2} + t ^{2 (^{'} + 1)} f ( t ) \frac{d}{d t} ( f ( t ) )}{f ( t ) ^{2} + g ( t ) ^{2} + t ^{2 (^{'} + 1)} f ( t ) ^{2}}

= \frac{f ( t ) \frac{d}{d t} ( f ( t ) ) + g ( t ) \frac{d}{d t} ( g ( t ) ) + ( ^{'} + 1 ) t ^{2^{'} + 1} f ( t ) ^{2} + t ^{2 (^{'} + 1)} f ( t ) \frac{d}{d t} ( f ( t ) )}{f ( t ) ^{2} + g ( t ) ^{2} + t ^{2 (^{'} + 1)} f ( t ) ^{2}}