What is the d/(dM)(M^2(C/2-M/3)) ?

The d/(dM)(M^2(C/2-M/3)) is MC-M^2

What is the first d/(dM)(M^2(C/2-M/3)) ?

The first d/(dM)(M^2(C/2-M/3)) is MC-M^2

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\frac{d}{d M} (M^{2} (\frac{C}{2} - \frac{M}{3}))

MC - M^{2}

Solution steps

\frac{d}{d M} (M^{2} (\frac{C}{2} - \frac{M}{3}))

Treat

C

as a constant

Apply the Product Rule:

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

f = M^{2}, g = \frac{C}{2} - \frac{M}{3}

= \frac{d}{d M} (M^{2}) (\frac{C}{2} - \frac{M}{3}) + \frac{d}{d M} (\frac{C}{2} - \frac{M}{3}) M^{2}

\frac{d}{d M} (M^{2}) = 2 M

\frac{d}{d M} (\frac{C}{2} - \frac{M}{3}) = - \frac{1}{3}

= 2 M (\frac{C}{2} - \frac{M}{3}) + (- \frac{1}{3}) M^{2}

Simplify

2 M (\frac{C}{2} - \frac{M}{3}) + (- \frac{1}{3}) M^{2} : MC - M^{2}

= MC - M^{2}

What is the d/(dM)(M^2(C/2-M/3)) ?
The d/(dM)(M^2(C/2-M/3)) is MC-M^2
What is the first d/(dM)(M^2(C/2-M/3)) ?
The first d/(dM)(M^2(C/2-M/3)) is MC-M^2