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If
2xy-2y=-3x^(3) then find
(dy)/(dx) in terms of
x and
y.
Answer:
(dy)/(dx)=

If $2 x y-2 y=-3 x^{3}$ then find $\frac{d y}{d x}$ in terms of $x$ and $y$ . $\newline$ Answer: $\frac{d y}{d x}=$

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Evaluate an exponential function

Full solution

Q. If

2 x y-2 y=-3 x^{3}

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 $2xy - 2y = -3x^3$ , and we need to find the derivative of $y$ with respect to $x$ , which is 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 $x$ , treating $y$ as a function of $x$ .
Implicit Differentiation: First, we differentiate the left side of the equation with respect to $x$ . The left side is $2xy - 2y$ . We apply the product rule to the term $2xy$ , which states that the derivative of a product $uv$ is $u'v + uv'$ . Here, $u = x$ and $v = y$ , so we get $2(y + xy')$ . We also differentiate $-2y$ with respect to $x$ , which gives us $2xy - 2y$ $0$ because $2xy - 2y$ $1$ is a function of $x$ .
Differentiate Left Side: Now, we differentiate the right side of the equation, which is $-3x^3$ . The derivative of $-3x^3$ with respect to $x$ is $-9x^2$ .
Differentiate Right Side: We now have the equation from the derivatives of both sides: $2(y + xy') - 2y' = -9x^2$ . We need to solve this equation for $y'$ to find $(dy/dx)$ .
Solve for $y'$ : Let's simplify the equation and collect all terms involving $y'$ on one side. We distribute the $2$ on the left side to get $2y + 2xy' - 2y' = -9x^2$ . Then we combine like terms to get $2y + (2x - 2)y' = -9x^2$ .
Isolate $y'$ : Next, we isolate the term involving $y'$ by subtracting $2y$ from both sides: $(2x - 2)y' = -9x^2 - 2y$ .
Final Solution: Now, we solve for $y'$ by dividing both sides by $(2x - 2)$ : $y' = \frac{-9x^2 - 2y}{2x - 2}$ .

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