Solve P = P0 - (1/\beta ) * ln(V/(V M))

Evaluate

P = P \times 0 - \frac{1}{β} \times ln (\frac{V}{V M})

Simplify

More Steps

P = - \frac{ln ( \frac{1}{M} )}{β}

Find the first partial derivative by treating the variable β as a constant and differentiating with respect to M

\frac{\partial P}{\partial M} = \frac{\partial}{\partial M} (- \frac{ln ( \frac{1}{M} )}{β})

Use differentiation rule \frac{\partial}{\partial x} (\frac{f ( x )}{g ( x )}) = \frac{\frac{\partial}{\partial x} ( f ( x )) \times g ( x ) - f ( x ) \times \frac{\partial}{\partial x} ( g ( x ) )}{( g ( x ) ) ^{2}}

\frac{\partial P}{\partial M} = - \frac{\frac{\partial}{\partial M} ( ln ( \frac{1}{M} ) ) β - ln ( \frac{1}{M} ) \times \frac{\partial}{\partial M} ( β )}{β ^{2}}

Evaluate

More Steps

\frac{\partial P}{\partial M} = - \frac{- \frac{1}{M} \times β - ln ( \frac{1}{M} ) \times \frac{\partial}{\partial M} ( β )}{β ^{2}}

Use \frac{\partial}{\partial x} (c) = 0 to find derivative

\frac{\partial P}{\partial M} = - \frac{- \frac{1}{M} \times β - ln ( \frac{1}{M} ) \times 0}{β ^{2}}

Evaluate

\frac{\partial P}{\partial M} = - \frac{- \frac{β}{M} - ln ( \frac{1}{M} ) \times 0}{β ^{2}}

Evaluate

More Steps

\frac{\partial P}{\partial M} = - \frac{- \frac{β}{M} - 0}{β ^{2}}

Removing 0 doesn't change the value,so remove it from the expression

\frac{\partial P}{\partial M} = - \frac{- \frac{β}{M}}{β ^{2}}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

\frac{\partial P}{\partial M} = \frac{\frac{β}{M}}{β ^{2}}

Solution

More Steps

\frac{\partial P}{\partial M} = \frac{1}{Mβ}

P = p0 (1/\beta ) * ln(v/(v m))