what is the second derivative of \frac{-12y}{12x+2y} Calculate *𝑑𝑦𝑑𝑥*dydx and *𝑑*2*𝑦𝑑𝑥*2d2ydx2 for 12*𝑥𝑦*+*𝑦*2=5. implicit

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 \( \frac{dy}{dx} \).

what is the second derivative of \frac{-12y}{12x+2y}? Calculate *𝑑𝑦/𝑑𝑥* and *𝑑^2𝑦/𝑑𝑥^2* for 12𝑥𝑦 + 𝑦^2=5. (implicit differentiation)

Learn how to calculate the second derivative using implicit differentiation for the equation 12xy + y^2 = 5. This detailed solution applies the quotient rule and explains the step-by-step process of finding both first and second derivatives. Ideal for students in advanced calculus classes (Grades 11-12).

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