To find the second derivative $\frac{d^2 y}{dx^2}$, we need to perform implicit differentiation. Let’s break down the steps.

Step 1: Given equation

We are given the equation:

$12xy + y^2 = 5.$

Step 2: First derivative using implicit differentiation

We differentiate both sides of the equation with respect to $x$:

$\frac{d}{dx}(12xy + y^2) = \frac{d}{dx}(5).$

Differentiating term-by-term:

$12 \cdot \left( x \frac{dy}{dx} + y \cdot 1 \right) + 2y \frac{dy}{dx} = 0.$

Expanding this results in:

$12y + 12x \frac{dy}{dx} + 2y \frac{dy}{dx} = 0.$

Now, we group all terms involving $\frac{dy}{dx}$:

$12x \frac{dy}{dx} + 2y \frac{dy}{dx} = -12y.$

Factor out $\frac{dy}{dx}$:

$\frac{dy}{dx} (12x + 2y) = -12y.$

Thus, the first derivative $\frac{dy}{dx}$ is:

$\frac{dy}{dx} = \frac{-12y}{12x + 2y}.$

Step 3: Second derivative $\frac{d^2 y}{dx^2}$

To find $\frac{d^2 y}{dx^2}$, we differentiate $\frac{dy}{dx}$ again with respect to $x$:

$\frac{d}{dx} \left( \frac{-12y}{12x + 2y} \right).$

We will apply the quotient rule:

$\frac{d^2 y}{dx^2} = \frac{ \left( (12x + 2y) \cdot \frac{d}{dx}(-12y) - (-12y) \cdot \frac{d}{dx}(12x + 2y) \right) }{ (12x + 2y)^2 }.$

Now, calculate each derivative separately:

1. $\frac{d}{dx}(-12y) = -12 \frac{dy}{dx}$.
2. $\frac{d}{dx}(12x + 2y) = 12 + 2 \frac{dy}{dx}$.

Plugging these into the quotient rule expression:

$\frac{d^2 y}{dx^2} = \frac{ (12x + 2y) (-12 \frac{dy}{dx}) - (-12y)(12 + 2 \frac{dy}{dx}) }{ (12x + 2y)^2 }.$

Expanding the numerator:

$\frac{d^2 y}{dx^2} = \frac{ -12 (12x + 2y) \frac{dy}{dx} + 12 \cdot 12y + 2(-12y) \frac{dy}{dx} }{ (12x + 2y)^2 }.$

Rearranging and simplifying:

$\frac{d^2 y}{dx^2} = \frac{ -12 (12x + 2y - 2y) \frac{dy}{dx} + 144y }{ (12x + 2y)^2 }.$

Final answer

$\frac{d^2 y}{dx^2} = \frac{ -12 \cdot 12x \cdot \frac{dy}{dx} + 144y }{ (12x + 2y)^2 }.$

Do you want more details or have any questions?

Related Questions

1. What are the critical points of the given equation using implicit differentiation?
2. How does implicit differentiation help find higher-order derivatives?
3. How does the presence of $x$ and $y$ in the same terms affect the derivative calculation?
4. How would the results differ if the given equation involved trigonometric functions?
5. What real-world situations might be modeled by an equation like $12xy + y^2 = 5$?

Tip

When differentiating implicitly, remember that each derivative involving $y$ should be multiplied by $\frac{dy}{dx}$.