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

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

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Evaluate a nonlinear function

Full solution

Q. If

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

then find

\frac{d y}{d x}

at the point

(1,2)

.

\newline

Answer:

\left.\frac{d y}{d x}\right|_{(1,2)}=

Differentiate and Solve: First, we need to differentiate both sides of the equation with respect to $x$ to find $\frac{dy}{dx}$ . $\newline$ Given equation: $y^2 = 2x^2 + y$ $\newline$ Differentiating both sides with respect to $x$ , we get: $\newline$ $\frac{d}{dx}(y^2) = \frac{d}{dx}(2x^2) + \frac{d}{dx}(y)$ $\newline$ Using the chain rule for $\frac{d}{dx}(y^2)$ , power rule for $\frac{d}{dx}(2x^2)$ , and remembering to apply the chain rule for $\frac{d}{dx}(y)$ since $y$ is a function of $x$ , we have: $\newline$ $\frac{dy}{dx}$ $0$
Isolate $\frac{dy}{dx}$ : Now, we need to solve for $\frac{dy}{dx}$ . $\newline$ Rearrange the terms to isolate $\frac{dy}{dx}$ on one side: $\newline$ $2y \cdot \frac{dy}{dx} - \frac{dy}{dx} = 4x$ $\newline$ Factor out $\frac{dy}{dx}$ : $\newline$ $\frac{dy}{dx} \cdot (2y - 1) = 4x$ $\newline$ Divide both sides by $(2y - 1)$ to solve for $\frac{dy}{dx}$ : $\newline$ $\frac{dy}{dx} = \frac{4x}{2y - 1}$
Substitute and Simplify: Next, we substitute the given point $(1,2)$ into the equation to find the specific value of $\frac{dy}{dx}$ at that point. $\newline$ Substitute $x = 1$ and $y = 2$ into the equation: $\newline$ $\frac{dy}{dx} = \frac{4(1)}{(2(2) - 1)}$ $\newline$ Simplify the equation: $\newline$ $\frac{dy}{dx} = \frac{4}{(4 - 1)}$ $\newline$ $\frac{dy}{dx} = \frac{4}{3}$

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