What is the d/(dt)(xy(t){z}(t)) ?

The d/(dt)(xy(t){z}(t)) is (dx}{dt)/y (t){z}(t)+(d/(dt)({y}(t)){z}(t)+d/(dt)({z}(t)){y}(t))x

What is the first d/(dt)(xy(t){z}(t)) ?

The first d/(dt)(xy(t){z}(t)) is (dx}{dt)/y (t){z}(t)+(d/(dt)({y}(t)){z}(t)+d/(dt)({z}(t)){y}(t))x

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\frac{d}{d t} (x y (t) z (t))

\frac{d x}{d t} y (t) z (t) + (\frac{d}{d t} (y (t)) z (t) + \frac{d}{d t} (z (t)) y (t)) x

Solution steps

\frac{d}{d t} (x y (t) z (t))

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = y (t) z (t)

= \frac{d x}{d t} y (t) z (t) + \frac{d}{d t} (y (t) z (t)) x

\frac{d}{d t} (y (t) z (t)) = \frac{d}{d t} (y (t)) z (t) + \frac{d}{d t} (z (t)) y (t)

= \frac{d x}{d t} y (t) z (t) + (\frac{d}{d t} (y (t)) z (t) + \frac{d}{d t} (z (t)) y (t)) x

What is the d/(dt)(xy(t){z}(t)) ?
The d/(dt)(xy(t){z}(t)) is (dx}{dt)/y (t){z}(t)+(d/(dt)({y}(t)){z}(t)+d/(dt)({z}(t)){y}(t))x
What is the first d/(dt)(xy(t){z}(t)) ?
The first d/(dt)(xy(t){z}(t)) is (dx}{dt)/y (t){z}(t)+(d/(dt)({y}(t)){z}(t)+d/(dt)({z}(t)){y}(t))x