Question

a. use implicit differentiation to find $\frac{dy}{dx}$. b. find the slope of the curve at the given point. $y^{3}=243x; (3,9)$ a. $\frac{dy}{dx}=square$

Explanation:

Step1: Differentiate both sides

Differentiate $y^{3}=243x$ with respect to $x$. Using the chain - rule on the left - hand side ($\frac{d}{dx}(y^{3}) = 3y^{2}\frac{dy}{dx}$) and the power - rule on the right - hand side ($\frac{d}{dx}(243x)=243$). So, $3y^{2}\frac{dy}{dx}=243$.

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

Divide both sides of the equation $3y^{2}\frac{dy}{dx}=243$ by $3y^{2}$ to get $\frac{dy}{dx}=\frac{243}{3y^{2}}=\frac{81}{y^{2}}$.

Step3: Find the slope at the given point

Substitute $x = 3$ and $y = 9$ into $\frac{dy}{dx}$. When $y = 9$, $\frac{dy}{dx}=\frac{81}{9^{2}}=\frac{81}{81}=1$.

Answer:

a. $\frac{dy}{dx}=\frac{81}{y^{2}}$
b. $1$