The differentiable functions x and y are related by the following equation: sin(y)=-5x Also, (dy)/(dt)=10. Find (dx)/(dt) when y=-pi.

Given Information: We are given that sin(y) = -5x and we need to find (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) * (dy)/(dt) because of the chain rule.Differentiate -5x: Differentiating -5x with respect to t gives us -5 * (dx)/(dt) because x is a function of t.Equating Derivatives: Now we equate the derivatives from both sides of the equation: cos(y) * (dy)/(dt) = -5 * (dx)/(dt)Substitute Given Value: We are given that (dy)/(dt) = 10. We substitute this value into the equation: cos(y) * 10 = -5 * (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: -1 * 10 = -5 * (dx)/(dt)Simplify Equation: Simplify the equation to solve for (dx)/(dt): -10 = -5 * (dx)/(dt)Isolate (dx)/(dt): Divide both sides by -5 to isolate (dx)/(dt): (dx)/(dt) = -10 / -5Final Value of (dx)/(dt): Simplify the fraction to get the final value of (dx)/(dt): (dx)/(dt) = 2

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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 $(\frac{dx}{dt})$ : $\frac{dx}{dt} = 2$