# Find the partial derivatives of m = ln(qh-2h^ 2)+2e(q-h^2+3)^4-7 ?

Dec 21, 2016

$\frac{\partial m}{\partial q} = \frac{1}{q - 2 h} + 8 e {\left(q - {h}^{2} + 3\right)}^{3}$

$\frac{\partial m}{\partial h} = \frac{q - 4 h}{q h - 2 {h}^{2}} - 16 e h {\left(q - {h}^{2} + 3\right)}^{3}$

#### Explanation:

If

$m = \ln \left(q h - 2 {h}^{2}\right) + 2 e {\left(q - {h}^{2} + 3\right)}^{4} - 7$,

then (assuming that $q$ and $h$ are variables) the partial derivatives are:

By treating $h$ as constant we get:

$\setminus \setminus \setminus \setminus \setminus \frac{\partial m}{\partial q} = \frac{1}{q h - 2 {h}^{2}} \cdot h + 2 e \cdot 4 {\left(q - {h}^{2} + 3\right)}^{3} \cdot 1$
$\therefore \frac{\partial m}{\partial q} = \frac{1}{q - 2 h} + 8 e {\left(q - {h}^{2} + 3\right)}^{3}$

And by treating $q$ as constant we get:

$\setminus \setminus \setminus \setminus \setminus \frac{\partial m}{\partial h} = \frac{1}{q h - 2 {h}^{2}} \cdot \left(q - 4 h\right) + 2 e \cdot 4 {\left(q - {h}^{2} + 3\right)}^{3} \cdot \left(- 2 h\right)$
$\therefore \frac{\partial m}{\partial h} = \frac{q - 4 h}{q h - 2 {h}^{2}} - 16 e h {\left(q - {h}^{2} + 3\right)}^{3}$