The differentiable functions x and y are related by the following equation: (1)/(y)=8-x Also, (dy)/(dt)=-0.5. Find (dx)/(dt) when y=0.2.

Differentiate with respect to t: First, we need to differentiate the given equation with respect to t. The equation is (1)/(y) = 8 - x. We will use implicit differentiation.Apply chain rule: Differentiating both sides of the equation with respect to t, we get: (d/dt)(1/y) = (d/dt)(8 - x) Since 8 is a constant, its derivative is 0, and the derivative of -x with respect to t is -(dx/dt).Substitute derivatives: The left side of the equation involves the derivative of a reciprocal function. Using the chain rule, the derivative of 1/y with respect to t is: (d/dt)(1/y) = -(dy/dt)/y^2Given values substitution: Substituting the derivatives into the differentiated equation, we get: -(dy/dt)/y^2 = -(dx/dt)Simplify to solve: We are given that (dy/dt) = -0.5 and we need to find (dx/dt) when y = 0.2. Let's substitute these values into the equation: -(-0.5)/(0.2^2) = -(dx/dt)Calculate final value: Simplify the equation to solve for (dx/dt): (dx/dt) = -(-0.5)/(0.2^2) (dx/dt) = 0.5/(0.04)Calculate the value of (dx/dt): (dx/dt) = 0.5/0.04 (dx/dt) = 12.5

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The differentiable functions
x and
y are related by the following equation:

(1)/(y)=8-x
Also,
(dy)/(dt)=-0.5.
Find
(dx)/(dt) when
y=0.2.

The differentiable functions $x$ and $y$ are related by the following equation: $\newline$ $\frac{1}{y}=8-x$ $\newline$ Also, $\frac{d y}{d t}=-0.5$ . $\newline$ Find $\frac{d x}{d t}$ when $y=0.2$ .

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Algebra 2

Simplify expressions using trigonometric identities

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Q. The differentiable functions

x

and

y

are related by the following equation:

\newline

\frac{1}{y}=8-x

\newline

Also,

\frac{d y}{d t}=-0.5

\newline

Find

\frac{d x}{d t}

when

y=0.2

Differentiate with respect to $t$ : First, we need to differentiate the given equation with respect to $t$ . The equation is $\frac{1}{y} = 8 - x$ . We will use implicit differentiation.
Apply chain rule: Differentiating both sides of the equation with respect to $t$ , we get: $\newline$ $(\frac{d}{dt})(\frac{1}{y}) = (\frac{d}{dt})(8 - x)$ $\newline$ Since $8$ is a constant, its derivative is $0$ , and the derivative of $-x$ with respect to $t$ is $-(\frac{dx}{dt})$ .
Substitute derivatives: The left side of the equation involves the derivative of a reciprocal function. Using the chain rule, the derivative of $\frac{1}{y}$ with respect to $t$ is: $\frac{d}{dt}(\frac{1}{y}) = -\frac{dy}{dt}/y^{2}$
Given values substitution: Substituting the derivatives into the differentiated equation, we get: $\newline$ $-\frac{dy}{dt}/y^2 = -\frac{dx}{dt}$
Simplify to solve: We are given that $\frac{dy}{dt} = -0.5$ and we need to find $\frac{dx}{dt}$ when $y = 0.2$ . Let's substitute these values into the equation: $\newline$ $-\left(-0.5\right)/\left(0.2^2\right) = -\left(\frac{dx}{dt}\right)$
Calculate final value: Simplify the equation to solve for $(\frac{dx}{dt}):$ $(\frac{dx}{dt}) = -(-\frac{0.5}{0.2^2})$ $(\frac{dx}{dt}) = \frac{0.5}{0.04}$
Calculate final value: Simplify the equation to solve for $(\frac{dx}{dt}):$ $(\frac{dx}{dt}) = -(-0.5)/(0.2^2)$ $(\frac{dx}{dt}) = \frac{0.5}{0.04}$ Calculate the value of $(\frac{dx}{dt}):$ $(\frac{dx}{dt}) = \frac{0.5}{0.04}$ $(\frac{dx}{dt}) = 12.5$