Question

a. use implicit differentiation to find \\(\frac{dy}{dx}). b. find the slope of the curve at the given point. \\(\cos 5y=x, (0,\frac{\pi}{10})\\) a. \\(\frac{dy}{dx}=\square\\)

Explanation:

Step1: Differentiate both sides of $\cos(5y)=x$ with respect to $x$

Using the chain - rule, the derivative of the left - hand side is $-\sin(5y)\cdot5\frac{dy}{dx}$, and the derivative of the right - hand side is $1$. So, $- 5\sin(5y)\frac{dy}{dx}=1$.

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

We get $\frac{dy}{dx}=-\frac{1}{5\sin(5y)}$.

Step3: Evaluate $\frac{dy}{dx}$ at the point $(0,\frac{\pi}{10})$

Substitute $y = \frac{\pi}{10}$ into $\frac{dy}{dx}=-\frac{1}{5\sin(5y)}$. Since $\sin(5\cdot\frac{\pi}{10})=\sin(\frac{\pi}{2}) = 1$, then $\frac{dy}{dx}=-\frac{1}{5\times1}=-\frac{1}{5}$.

Answer:

a. $\frac{dy}{dx}=-\frac{1}{5\sin(5y)}$
b. The slope of the curve at the point $(0,\frac{\pi}{10})$ is $-\frac{1}{5}$