Question

If 2x^2 + y^2 = 17 then evaluate the second derivative of y with respect to x when x = 2 and y = 3. Round your answer to 2 decimal places. Use the hyphen symbol, -, for negative values.

Answer 1

Differentiating both sides with respect to
`x` gives

`(\mathrm d)/(\mathrm dx)[2x^2+y^2]=(\mathrm d)/(\mathrm dx)17`

`4x+2y(\mathrm dy)/(\mathrm dx)=0`

`(\mathrm dy)/(\mathrm dx)=-(4x)/(2y)=-\frac{2x}y`

Differentiating again, we get

`(\mathrm d^2y)/(\mathrm dx^2)=-(2y-2x(\mathrm dy)/(\mathrm dx))/(y^2)`

`(\mathrm d^2y)/(\mathrm dx^2)=\frac{2x\left(-\frac{2x}y\right)-2y}{y^2}`

`(\mathrm d^2y)/(\mathrm dx^2)=-(4x^2-2y^2)/(y^3)`

Plug in
`x=2` and
`y=3` and you're done.