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

Differentiate with respect to x: Differentiate both sides of the equation with respect to x. We have y^3 - x^2 = -1. To find dy/dx, we need to differentiate both sides of the equation with respect to x, using the chain rule for the y^3 term since y is a function of x. d/dx(y^3) - d/dx(x^2) = d/dx(-1) 3y^2(dy/dx) - 2x = 0Solve for dy/dx: Solve for dy/dx. Now we isolate dy/dx on one side of the equation. 3y^2(dy/dx) = 2x (dy/dx) = 2x / (3y^2)Substitute point into derivative: Substitute the point (3,2) into the derivative to find the slope at that point. We substitute x = 3 and y = 2 into the equation from Step 2. (dy/dx)|_((3,2)) = 2(3) / (3(2)^2) (dy/dx)|_((3,2)) = 6 / (3*4) (dy/dx)|_((3,2)) = 6 / 12 (dy/dx)|_((3,2)) = 1/2

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

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

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Algebra 1

Transformations of quadratic functions

Full solution

Q. If

y^{3}-x^{2}=-1

then find

\frac{d y}{d x}

at the point

(3,2)

\newline

Answer:

\left.\frac{d y}{d x}\right|_{(3,2)}=

Differentiate with respect to $x$ : Differentiate both sides of the equation with respect to $x$ . We have $y^3 - x^2 = -1$ . To find $\frac{dy}{dx}$ , we need to differentiate both sides of the equation with respect to $x$ , using the chain rule for the $y^3$ term since $y$ is a function of $x$ . $\frac{d}{dx}(y^3) - \frac{d}{dx}(x^2) = \frac{d}{dx}(-1)$ $3y^2(\frac{dy}{dx}) - 2x = 0$
Solve for $\frac{dy}{dx}$ : Solve for $\frac{dy}{dx}$ . Now we isolate $\frac{dy}{dx}$ on one side of the equation. $3y^2\left(\frac{dy}{dx}\right) = 2x$ $\frac{dy}{dx} = \frac{2x}{3y^2}$
Substitute point into derivative: Substitute the point $(3,2)$ into the derivative to find the slope at that point. $\newline$ We substitute $x = 3$ and $y = 2$ into the equation from Step $2$ . $\newline$ $(\frac{dy}{dx})|_{(3,2)} = \frac{2(3)}{(3(2)^2)}$ $\newline$ $(\frac{dy}{dx})|_{(3,2)} = \frac{6}{(3\cdot4)}$ $\newline$ $(\frac{dy}{dx})|_{(3,2)} = \frac{6}{12}$ $\newline$ $(\frac{dy}{dx})|_{(3,2)} = \frac{1}{2}$