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If
-2y-4x^(2)+xy-5=0 then find
(dy)/(dx) in terms of
x and
y

If $-2 y-4 x^{2}+x y-5=0$ then find $\frac{d y}{d x}$ in terms of $x$ and $y$

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Full solution

Q. If

-2 y-4 x^{2}+x y-5=0

then find

\frac{d y}{d x}

in terms of

x

and

y

Apply Implicit Differentiation: Given the equation $-2y - 4x^2 + xy - 5 = 0$ , 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 differentiating both sides of the equation with respect to $x$ while treating $y$ as a function of $x$ .
Differentiate Each Term: Differentiate each term of the equation with respect to $x$ . The derivative of $-2y$ with respect to $x$ is $-2\frac{dy}{dx}$ , since $y$ is a function of $x$ . The derivative of $-4x^2$ with respect to $x$ is $-8x$ . The derivative of $xy$ with respect to $x$ is $-2y$ $1$ by the product rule. The derivative of $-2y$ $2$ with respect to $x$ is $-2y$ $4$ , since it is a constant.
Collect and Rearrange Terms: Putting it all together, we have: $\newline$ $-2\frac{dy}{dx} - 8x + y + x\frac{dy}{dx} = 0$
Factor Out $(\frac{dy}{dx})$ : Now, we need to solve for $(\frac{dy}{dx})$ . To do this, we will collect all the terms involving $(\frac{dy}{dx})$ on one side and the remaining terms on the other side.
Divide to Solve for $(dy)/(dx)$ : Combine like terms and rearrange the equation: $\newline$ $-2\frac{dy}{dx} + x\frac{dy}{dx} = 8x - y$
Final Answer: Factor out $(\frac{dy}{dx})$ from the left side of the equation: $\newline$ (\frac{dy}{dx})(\(-2 + x) = $8$ x - y
Final Answer: Factor out $(\frac{dy}{dx})$ from the left side of the equation: $\newline$ $(\frac{dy}{dx})(-2 + x) = 8x - y$ Finally, divide both sides by $(-2 + x)$ to solve for $(\frac{dy}{dx}):$ $\newline$ $(\frac{dy}{dx}) = \frac{8x - y}{-2 + x}$
Final Answer: Factor out $(dy)/(dx)$ from the left side of the equation: $\newline$ $(dy)/(dx)(-2 + x) = 8x - y$ Finally, divide both sides by $(-2 + x)$ to solve for $(dy)/(dx)$ : $\newline$ $(dy)/(dx) = (8x - y) / (-2 + x)$ We have found the derivative of y with respect to x in terms of x and y. The final answer is: $\newline$ $(dy)/(dx) = (8x - y) / (-2 + x)$

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