Question

Find the value of d^2y/(dx^2) for the hyperbola defined by the equation y^2-x^2=7 at the point (3,4).
I know that the first derivative equation is x/y and is equals 3/4 at the point of (3,4) but I do not know how to find the second derivative.

Answer 1

Find the second derivative of
\[
f(x)=\sqrt{x^2+7}
\]

Answer 2

\[
f'(x)=\frac{x}{\sqrt{x^2+7}}
\]

Answer 3

\[
f''(x)=\frac{7}{\left(x^2+7\right)^{3/2}}
\]

Answer 4

\[
f''(3)=\frac 7{64}
\]

Answer 5

You can also do it by implicit differentiation

Answer 6

First you find
\[
y'(x)=\frac{x}{y(x)}
\]

Answer 7

then you find
\[
y''(x)=\frac{y(x)-x y'(x)}{y(x)^2}
\]

Answer 8

Replace x by 3, y'(x) by 3/4, y(x) by 4 you obtain the same result

Answer 9

Okay so would that look like 4-3(3/4)/16^2? @eliesaab