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

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

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Math Problems

Grade 8

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: $\newline$ $\frac{dy}{dx} = \frac{4}{(4 - 1)}$ $\newline$ $\frac{dy}{dx} = \frac{4}{3}$