Full solution

Q. If

0=5 x^{2}+5 y+x^{3} y

then find

\frac{d y}{d x}

in terms of

x

and

y

\newline

Answer:

\frac{d y}{d x}=

Given Equation: We are given the equation $0 = 5x^{2} + 5y + x^{3}y$ , and we need to find the derivative of $y$ with respect to $x$ , denoted as $\frac{dy}{dx}$ . To do this, we will use implicit differentiation, which involves taking the derivative of both sides of the equation with respect to $x$ , while treating $y$ as a function of $x$ .
Differentiate $5x^{2}$ : First, we differentiate the term $5x^{2}$ with respect to $x$ . The derivative of $5x^{2}$ with respect to $x$ is $10x$ .
Differentiate $5y$ : Next, we differentiate the term $5y$ with respect to $x$ . Since $y$ is a function of $x$ , we apply the chain rule and get $5\frac{dy}{dx}$ .
Differentiate $x^{3}y$ : Now, we differentiate the term $x^{3}y$ with respect to $x$ . This requires the product rule since it is the product of two functions of $x$ . The derivative of $x^{3}y$ with respect to $x$ is $3x^{2}y + x^{3}(\frac{dy}{dx})$ .
Combine Derivatives: Now we combine the derivatives we found in the previous steps and set the derivative of the entire left side of the equation equal to the derivative of the right side, which is $0$ . This gives us the equation $10x + 5\frac{dy}{dx} + 3x^{2}y + x^{3}\frac{dy}{dx} = 0$ .
Solve for (\frac{dy}{dx}): We need to solve for \((\frac{dy}{dx}), so we group the terms containing \((\frac{dy}{dx}) on one side and the rest on the other side. This gives us \(\(5(\frac{dy}{dx}) + x^{ $3$ }(\frac{dy}{dx}) = $-10$ x - $3$ x^{ $2$ }y.
Factor Out $(\frac{dy}{dx}):</b> Factor out \$(\frac{dy}{dx})$ from the left side of the equation to get $(\frac{dy}{dx})(5 + x^{3}) = -10x - 3x^{2}y$ .
Final Solution: Now, we solve for $\frac{dy}{dx}$ by dividing both sides of the equation by $(5 + x^{3})$ . This gives us $\frac{dy}{dx} = \frac{-10x - 3x^{2}y}{5 + x^{3}}$ .

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