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If
y^(2)+3x^(3)=2y then find
(dy)/(dx) at the point
(-1,3).
Answer:
(dy)/(dx)|_((-1,3))=

If $y^{2}+3 x^{3}=2 y$ then find $\frac{d y}{d x}$ at the point $(-1,3)$ . $\newline$ Answer: $\left.\frac{d y}{d x}\right|_{(-1,3)}=$

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Evaluate a nonlinear function

Full solution

Q. If

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

then find

\frac{d y}{d x}

at the point

(-1,3)

.

\newline

Answer:

\left.\frac{d y}{d x}\right|_{(-1,3)}=

Differentiate and Simplify: To find $\frac{dy}{dx}$ , we need to differentiate both sides of the equation with respect to $x$ , using implicit differentiation. $\newline$ The equation is $y^2 + 3x^3 = 2y$ . $\newline$ Differentiating both sides with respect to $x$ gives us: $\newline$ $\frac{d}{dx}(y^2) + \frac{d}{dx}(3x^3) = \frac{d}{dx}(2y)$ $\newline$ Using the chain rule for $\frac{d}{dx}(y^2)$ and the power rule for $\frac{d}{dx}(3x^3)$ and $\frac{d}{dx}(2y)$ , we get: $\newline$ $2y\frac{dy}{dx} + 9x^2 = 2\frac{dy}{dx}$
Isolate $\frac{dy}{dx}$ : Now we need to solve for $\frac{dy}{dx}$ . We can rearrange the terms to isolate $\frac{dy}{dx}$ on one side: $2y\left(\frac{dy}{dx}\right) - 2\left(\frac{dy}{dx}\right) = -9x^2$ Factor out $\frac{dy}{dx}$ : $\left(\frac{dy}{dx}\right)(2y - 2) = -9x^2$ Now, divide both sides by $(2y - 2)$ to solve for $\frac{dy}{dx}$ : $\frac{dy}{dx} = \frac{-9x^2}{2y - 2}$
Evaluate at $(-1,3)$ : We need to evaluate $\frac{dy}{dx}$ at the point $(-1,3)$ . $\newline$ Substitute $x = -1$ and $y = 3$ into the expression for $\frac{dy}{dx}$ : $\newline$ $\left(\frac{dy}{dx}\right)|_{(-1,3)} = -9(-1)^2 / (2(3) - 2)$ $\newline$ Calculate the numerator and the denominator separately: $\newline$ Numerator: $-9(-1)^2 = -9$ $\newline$ Denominator: $2(3) - 2 = 6 - 2 = 4$ $\newline$ Now, divide the numerator by the denominator: $\newline$ $\left(\frac{dy}{dx}\right)|_{(-1,3)} = -9 / 4$

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