What is the d/(dt)(({P}(t,M))(r(t,M){P}(t,M)(M-{P}(t,M)))) ?

The d/(dt)(({P}(t,M))(r(t,M){P}(t,M)(M-{P}(t,M)))) is

What is the first d/(dt)(({P}(t,M))(r(t,M){P}(t,M)(M-{P}(t,M)))) ?

The first d/(dt)(({P}(t,M))(r(t,M){P}(t,M)(M-{P}(t,M)))) is

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\frac{d}{d t} ((P (t, M)) (r (t, M) P (t, M) (M - P (t, M))))

\frac{d}{d t} (P (t, M)) r (t, M) P (t, M) (M - P (t, M)) + (\frac{d}{d t} (r (t, M)) P (t, M) (M - P (t, M)) + (\frac{d}{d t} (P (t, M)) (M - P (t, M)) + (0 - \frac{d}{d t} (P (t, M))) P (t, M)) r (t, M)) P (t, M)

Solution steps

\frac{d}{d t} ((P (t, M)) (r (t, M) P (t, M) (M - P (t, M))))

Treat

M

as a constant

Apply the Product Rule:

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

f = P (t, M), g = r (t, M) P (t, M) (M - P (t, M))

= \frac{d}{d t} (P (t, M)) r (t, M) P (t, M) (M - P (t, M)) + \frac{d}{d t} (r (t, M) P (t, M) (M - P (t, M))) P (t, M)

\frac{d}{d t} (r (t, M) P (t, M) (M - P (t, M))) = \frac{d}{d t} (r (t, M)) P (t, M) (M - P (t, M)) + (\frac{d}{d t} (P (t, M)) (M - P (t, M)) + (0 - \frac{d}{d t} (P (t, M))) P (t, M)) r (t, M)

= \frac{d}{d t} (P (t, M)) r (t, M) P (t, M) (M - P (t, M)) + (\frac{d}{d t} (r (t, M)) P (t, M) (M - P (t, M)) + (\frac{d}{d t} (P (t, M)) (M - P (t, M)) + (0 - \frac{d}{d t} (P (t, M))) P (t, M)) r (t, M)) P (t, M)

(2 x - 1) \cdot lo g_{10} (5)

x (x + 1) - (4 - x) (x + 1 = 6 (4 - x))

15 n^{2} + n (lo g_{2} (n) + 1)

4 (1 - x)^{- 1} - 3 (3 + 2 x)^{- 1}

\frac{1}{28} (2 ln (2) + 1)

What is the d/(dt)(({P}(t,M))(r(t,M){P}(t,M)(M-{P}(t,M)))) ?
The d/(dt)(({P}(t,M))(r(t,M){P}(t,M)(M-{P}(t,M)))) is
What is the first d/(dt)(({P}(t,M))(r(t,M){P}(t,M)(M-{P}(t,M)))) ?
The first d/(dt)(({P}(t,M))(r(t,M){P}(t,M)(M-{P}(t,M)))) is