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

If $-x-x y=-y^{3}$ 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

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

then find

\frac{d y}{d x}

in terms of

x

and

y

.

\newline

Answer:

\frac{d y}{d x}=

Differentiate with respect to $x$ : We are given the equation $-x - xy = -y^3$ . To find $\frac{dy}{dx}$ , we need to differentiate both sides of the equation with respect to $x$ , treating $y$ as a function of $x$ (implicit differentiation). $\newline$ Differentiate both sides of the equation with respect to $x$ : $\newline$ $\frac{d}{dx}(-x - xy) = \frac{d}{dx}(-y^3)$ $\newline$ This gives us: $\newline$ $-1 - (x \cdot \frac{d}{dx}(y) + y \cdot \frac{d}{dx}(x)) = -3y^2 \cdot \frac{d}{dx}(y)$ $\newline$ Simplify the differentiation: $\newline$ $-1 - (x \cdot \frac{dy}{dx} + y \cdot 1) = -3y^2 \cdot \frac{dy}{dx}$
Simplify the expression: Now we have an equation with $\frac{dy}{dx}$ terms that we can solve for $\frac{dy}{dx}$ :
$-1 - x\frac{dy}{dx} - y = -3y^2\frac{dy}{dx}$
Rearrange the terms to isolate $\frac{dy}{dx}$ on one side:
$x\frac{dy}{dx} + 3y^2\frac{dy}{dx} = -1 + y$
Factor out $\frac{dy}{dx}$ from the left side:
$\frac{dy}{dx}(x + 3y^2) = -1 + y$
Isolate $\frac{dy}{dx}$ : Finally, divide both sides by $(x + 3y^2)$ to solve for $\frac{dy}{dx}$ : $\frac{dy}{dx} = \frac{-1 + y}{x + 3y^2}$ This is the derivative of $y$ with respect to $x$ in terms of $x$ and $y$ .

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