# How do you use implicit differentiation to find (d^2y)/(dx^2) given 4y^2+2=3x^2?

Mar 17, 2017

$\frac{{d}^{2} y}{{\mathrm{dx}}^{2}} = \frac{3}{8 y} \left[2 - \frac{3 {x}^{2}}{2 {y}^{2}}\right]$

#### Explanation:

Implicit differentiation is a special case of the chain rule for derivatives.

Generally differentiation problems involve functions i.e. $y = f \left(x\right)$ - written explicitly as functions of $x$. However, some functions y are written implicitly as functions of $x$.

So what we do is to treat $y$ as $y = y \left(x\right)$ and use chain rule. This means differentiating $y$ w.r.t. $y$, but as we have to derive w.r.t. $x$, as per chain rule, we multiply it by $\frac{\mathrm{dy}}{\mathrm{dx}}$.

Hence implicit differentiating $4 {y}^{2} + 2 = 3 {x}^{2}$, we get

$4 \times 2 y \times \frac{\mathrm{dy}}{\mathrm{dx}} + 0 = 3 \times 2 x$

or $8 y \frac{\mathrm{dy}}{\mathrm{dx}} = 6 x$ ........................(1)

and $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{6 x}{8 y} = \frac{3 x}{4 y}$ ........................(2)

Now implicitly differentiating (1) further, we get

$8 \frac{\mathrm{dy}}{\mathrm{dx}} \times \frac{\mathrm{dy}}{\mathrm{dx}} + 8 y \times \frac{{d}^{2} y}{{\mathrm{dx}}^{2}} = 6$

or using (2) we get $8 {\left(\frac{3 x}{4 y}\right)}^{2} + 8 y \frac{{d}^{2} y}{{\mathrm{dx}}^{2}} = 6$

or $8 \frac{9 {x}^{2}}{16 {y}^{2}} + 8 y \frac{{d}^{2} y}{{\mathrm{dx}}^{2}} = 6$

or $\frac{{d}^{2} y}{{\mathrm{dx}}^{2}} = \frac{1}{8 y} \left[6 - \frac{9 {x}^{2}}{2 {y}^{2}}\right] = \frac{3}{8 y} \left[2 - \frac{3 {x}^{2}}{2 {y}^{2}}\right]$