if 5x^2 + y^4 = -9 then evaluate (d^2(y))/(dx^2) when x = 2 and y = 1. round your answer to two decimal places. use the hyphen symbol, -, for negative values

(d means derivative)

This question involves implicit differentiation. To do this, memorise the rule that where y is a function of x...

To differentiate a function of y with respect to x, differentiate with respect to y and multiply by dy/dx. For example d/dx(y^2) = d/dy(y^2) * dy/dx = 2y * dy/dx.

So with the problem in hand:

5x^2 + y^4 = -9

Differentiating both sides with respect to x:

10x + (4y^3)*(dy/dx) = 0

Rearranging to solve for dy/dx:

dy/dx = -10x / 4y^3

Next we need to use the quotient rule to differentiate again. Let u = 10x and v = 4y^3:

du/dx = 10
dv/dx = 12y^2 * dy/dx

Using the quotient rule, where:

d/dx(u/v) = (v(du/dx) - u(dv/dx))/(v^2)

Hence:

(d^2(y))/(dx^2) = (40y^3 - (120y^2 * dy/dx))/(16y^6)

Then we can just substitute x = 2 and y = 1 into the formula for dy/dx, and consequently calculate the second derivative using the second formula. I don't have a calculator to hand but it's just a matter of subbing in the values.

I hope this helps you :)