In Exercises 37 and 38 , use implicit differentiation to
find the second derivative .
37.

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Mar 12,2025

Question

In Exercises 37 and 38 , use implicit differentiation to find the second derivative $$ \frac{d^{2} y}{d x^{2}} $$. 37. $$ x^{2}+3 y^{2}=5 $$

UpStudy AI · Accepted Answer

Differentiate the equation
$$
x^2 + 3y^2 = 5
$$
with respect to $$ x $$.

1. Differentiate both sides:
   $$
   \frac{d}{dx}(x^2) + \frac{d}{dx}(3y^2) = \frac{d}{dx}(5)
   $$
   This yields:
   $$
   2x + 6y \cdot \frac{dy}{dx} = 0
   $$

2. Solve for $$\frac{dy}{dx}$$:
   $$
   6y \cdot \frac{dy}{dx} = -2x \quad \Longrightarrow \quad \frac{dy}{dx} = -\frac{2x}{6y} = -\frac{x}{3y}
   $$

3. Differentiate $$\frac{dy}{dx} = -\frac{x}{3y}$$ with respect to $$ x $$ to find the second derivative:
   $$
   \frac{d^2y}{dx^2} = \frac{d}{dx}\left(-\frac{x}{3y}\right)
   $$
   Write this as:
   $$
   \frac{dy}{dx} = -\frac{1}{3} \cdot \frac{x}{y}
   $$

4. Use the quotient rule for $$\frac{x}{y}$$. Recall, the quotient rule is given by:
   $$
   \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}
   $$
   where $$ u = x $$ (so $$\frac{du}{dx} = 1$$) and $$ v = y $$ (so $$\frac{dv}{dx} = \frac{dy}{dx} = -\frac{x}{3y} $$). Therefore,
   $$
   \frac{d}{dx}\left(\frac{x}{y}\right) = \frac{y \cdot 1 - x \cdot \left(-\frac{x}{3y}\right)}{y^2} = \frac{y + \frac{x^2}{3y}}{y^2}
   $$

5. Simplify the expression by combining the terms in the numerator:
   $$
   y + \frac{x^2}{3y} = \frac{3y^2 + x^2}{3y}
   $$
   Thus, the derivative becomes:
   $$
   \frac{d}{dx}\left(\frac{x}{y}\right) = \frac{\frac{3y^2 + x^2}{3y}}{y^2} = \frac{3y^2 + x^2}{3y^3}
   $$

6. Reintroduce the constant factor $$-\frac{1}{3}$$:
   $$
   \frac{d^2y}{dx^2} = -\frac{1}{3} \cdot \frac{3y^2 + x^2}{3y^3} = -\frac{3y^2+x^2}{9y^3}
   $$

The final answer is:
$$
\frac{d^2y}{dx^2} = -\frac{3y^2+x^2}{9y^3}
$$
$$ \frac{d^{2} y}{d x^{2}} = -\frac{3y^{2} + x^{2}}{9y^{3}} $$

In Exercises 37 and 38 , use implicit differentiation to
find the second derivative .
37.

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In Exercises 37 and 38 , use implicit differentiation to find the second derivative . 37.