The differentiable functions $x$ and $y$ are related by the following equation: $\newline$ $\sin (y)=-5 x$ $\newline$ Also, $\frac{d y}{d t}=10$ . $\newline$ Find $\frac{d x}{d t}$ when $y=-\pi$ .

Full solution

Q. The differentiable functions

x

and

y

are related by the following equation:

\newline

\sin (y)=-5 x

\newline

Also,

\frac{d y}{d t}=10

\newline

Find

\frac{d x}{d t}

when

y=-\pi

Given Information: We are given that $\sin(y) = -5x$ and we need to find $\frac{dx}{dt}$ when $y = -\pi$ . To do this, we will differentiate both sides of the equation with respect to $t$ .
Differentiate $\sin(y)$ : Differentiating $\sin(y)$ with respect to $t$ gives us $\cos(y) \cdot \frac{dy}{dt}$ because of the chain rule.
Differentiate $-5x$ : Differentiating $-5x$ with respect to $t$ gives us $-5 \times \frac{dx}{dt}$ because $x$ is a function of $t$ .
Equating Derivatives: Now we equate the derivatives from both sides of the equation: $\newline$ $\cos(y) \cdot \frac{dy}{dt} = -5 \cdot \frac{dx}{dt}$
Substitute Given Value: We are given that $\frac{dy}{dt} = 10$ . We substitute this value into the equation: $\newline$ $\cos(y) \cdot 10 = -5 \cdot \frac{dx}{dt}$
Find $\cos(y)$ : We need to find the value of $\cos(y)$ when $y = -\pi$ . The cosine of $-\pi$ is $-1$ .
Substitute $\cos(y)$ : Substitute $\cos(y) = -1$ into the equation: $\newline$ $-1 \times 10 = -5 \times \frac{dx}{dt}$
Simplify Equation: Simplify the equation to solve for $(\frac{dx}{dt})$ : $\newline$ $-10 = -5 \times (\frac{dx}{dt})$
Isolate $(dx)/(dt)$ : Divide both sides by $-5$ to isolate $(dx)/(dt)$ : $\newline$ $(dx)/(dt) = -10 / -5$
Final Value of $(\frac{dx}{dt})$ : Simplify the fraction to get the final value of $(\frac{dx}{dt})$ : $\frac{dx}{dt} = 2$