Given that x 2 - y 2 = 25 then show d 2 y d x 2 = - 25 y 3 ?

Question

Given that             x      2        -          y      2        =    25   then show                               d          2                y                                          d            x                    2                      =    -          25              y        3            ?

Van Delgado · Accepted Answer

First, a note on notation: when solving equations, expressions are derived or demonstrated, not shown.
So we are asked to show that given:

x^{2} - y^{2} = 25

..... [A]

Then

\frac{d}{d x} (x^{2}) - \frac{d}{d x} (y^{2}) = \frac{d}{d x} (25)

We start by implicitly differentiating the initial equation [A]:

\frac{d}{d x} (x^{2}) - \frac{d}{d x} (y^{2}) = \frac{d}{d x} (25)

∴ 2 x - 2 y \frac{d y}{d x} = 0

∴ \frac{d y}{d x} = \frac{2 x}{2 y} = \frac{x}{y}

..... [B]

We now differentiate this second equation [B] implicitly and apply the quotient rule, giving:

\frac{d^{2} y}{{d x}^{2}} = \frac{(y) (\frac{d}{d x} x) - (x) (\frac{d}{d x} y)}{{(y)}^{2}}

= \frac{(y) (1) - (x) (\frac{d y}{d x})}{{(y)}^{2}}

= \frac{y - x (\frac{x}{y})}{y^{2}}

= \frac{(y^{2} - x^{2}) \frac{1}{y}}{y^{2}}

using [B]

= \frac{(y^{2} - x^{2}) \frac{1}{y}}{y^{2}}

= \frac{y^{2} - x^{2}}{y^{3}}

= - \frac{25}{y^{3}}

using [A] QED

Given that x^2-y^2=25 then show (d^2y)/(dx^2) =-25/y^3?

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