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.

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.

eliesaab · Answer

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

eliesaab · Answer

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

eliesaab · Answer

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

eliesaab · Answer

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

eliesaab · Answer

You can also do it by implicit differentiation

eliesaab · Answer

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

eliesaab · Answer

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

eliesaab · Answer

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

canada907cat · Answer

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