Question

a. use implicit differentiation to find $\frac{dy}{dx}$
b. find the slope of the curve at the given point.
$sin y = 3x^{3}-3,(1,pi)$

Explanation:

Step1: Differentiate both sides

Differentiate $\sin y = 3x^{3}-3$ with respect to $x$.
Using the chain - rule on the left - hand side, we have $\cos y\frac{dy}{dx}=9x^{2}$.

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

Isolate $\frac{dy}{dx}$ by dividing both sides by $\cos y$. So, $\frac{dy}{dx}=\frac{9x^{2}}{\cos y}$.

Step3: Find the slope at the given point

Substitute $x = 1$ and $y=\pi$ into $\frac{dy}{dx}$.
Since $\cos\pi=- 1$ and $x = 1$, then $\frac{dy}{dx}=\frac{9\times1^{2}}{\cos\pi}=\frac{9}{-1}=-9$.

Answer:

a. $\frac{dy}{dx}=\frac{9x^{2}}{\cos y}$
b. $-9$