differentiate implicitly to find dy/dx. then find the slope of the curve at the given point. y² - 2x³ = 9…

Explanation:

Step1: Differentiate both sides

Differentiate $y^{2}-2x^{3}=9$ with respect to $x$. Using the chain - rule for $y^{2}$ (where $\frac{d}{dx}(y^{2}) = 2y\frac{dy}{dx}$) and the power - rule for $2x^{3}$ ($\frac{d}{dx}(2x^{3})=6x^{2}$), we get $2y\frac{dy}{dx}-6x^{2}=0$.

Step2: Solve for $\frac{dy}{dx}$

Add $6x^{2}$ to both sides: $2y\frac{dy}{dx}=6x^{2}$. Then divide both sides by $2y$ to isolate $\frac{dy}{dx}$: $\frac{dy}{dx}=\frac{6x^{2}}{2y}=\frac{3x^{2}}{y}$.

Step3: Find the slope at the point

Substitute $x = 2$ and $y=-5$ into $\frac{dy}{dx}$. $\frac{dy}{dx}\big|_{(2,-5)}=\frac{3\times2^{2}}{-5}=\frac{3\times4}{-5}=-\frac{12}{5}$.

Answer:

$\frac{dy}{dx}=\frac{3x^{2}}{y}$, and the slope at the point $(2, - 5)$ is $-\frac{12}{5}$