Answer:
Step-by-step explanation:
We have the equation:
And we want to find d²y/dx² at the point (-2, -1).
So, let's take the derivative of both sides with respect to x:
On the left, let's implicitly differentiate:
Differentiate normally on the left:
Solve for the first derivative. Divide both sides by 4y:
Now, let's take the derivative of both sides again:
We will need to use the quotient rule:
So:
Differentiate:
Simplify:
Substitute x/2y for dy/dx. This yields:
Simplify:
Simplify. Multiply both the numerator and denominator by 2y. So:
Reduce. Therefore, our second derivative is:
We want to find the second derivative at the point (-2, -1).
So, let's substitute -2 for x and -1 for y. This yields:
Evaluate:
Multiply:
Subtract:
Reduce. So, our answer is:
And we're done!