If x^(2)-y^(3)-5=4x then find (dy)/(dx) at the point (-4,3). Answer: (dy)/(dx)|_((-4,3))=

Differentiate Equation: First, we need to find the derivative of the given equation with respect to x. The equation is x^2 - y^3 - 5 = 4x. We will use implicit differentiation because the equation involves both x and y. Differentiate both sides of the equation with respect to x: d/dx (x^2) - d/dx (y^3) - d/dx (5) = d/dx (4x) 2x - 3y^2 (dy/dx) - 0 = 4Solve for dy/dx: Now, we solve for dy/dx: 2x - 3y^2 (dy/dx) = 4 -3y^2 (dy/dx) = 4 - 2x (dy/dx) = (4 - 2x) / (-3y^2)Substitute Point for Slope: Next, we substitute the point (-4,3) into the derivative to find the slope at that point: (dy/dx)|_((-4,3)) = (4 - 2(-4)) / (-3(3)^2) (dy/dx)|_((-4,3)) = (4 + 8) / (-3 * 9) (dy/dx)|_((-4,3)) = 12 / (-27) (dy/dx)|_((-4,3)) = -4 / 9

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Algebra 2

Simplify variable expressions using properties

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Q. If

x^{2}-y^{3}-5=4 x

then find

\frac{d y}{d x}

at the point

(-4,3)

\newline

Answer:

\left.\frac{d y}{d x}\right|_{(-4,3)}=

Differentiate Equation: First, we need to find the derivative of the given equation with respect to $x$ . The equation is $x^2 - y^3 - 5 = 4x$ . We will use implicit differentiation because the equation involves both $x$ and $y$ . $\newline$ Differentiate both sides of the equation with respect to $x$ : $\newline$ $\frac{d}{dx} (x^2) - \frac{d}{dx} (y^3) - \frac{d}{dx} (5) = \frac{d}{dx} (4x)$ $\newline$ $2x - 3y^2 \frac{dy}{dx} - 0 = 4$
Solve for $\frac{dy}{dx}$ : Now, we solve for $\frac{dy}{dx}$ : $2x - 3y^2 \left(\frac{dy}{dx}\right) = 4$ $-3y^2 \left(\frac{dy}{dx}\right) = 4 - 2x$ $\frac{dy}{dx} = \frac{4 - 2x}{-3y^2}$
Substitute Point for Slope: Next, we substitute the point $(-4,3)$ into the derivative to find the slope at that point: $\newline$ $(\frac{dy}{dx})|_{(-4,3)} = \frac{(4 - 2(-4))}{(-3(3)^2)}$ $\newline$ $(\frac{dy}{dx})|_{(-4,3)} = \frac{(4 + 8)}{(-3 \times 9)}$ $\newline$ $(\frac{dy}{dx})|_{(-4,3)} = \frac{12}{(-27)}$ $\newline$ $(\frac{dy}{dx})|_{(-4,3)} = -\frac{4}{9}$