The differentiable functions x and y are related by the following equation: sin(x)+cos(y)=sqrt2 Also, (dx)/(dt)=5. Find (dy)/(dt) when y=(pi)/(4) and 0

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The differentiable functions  x and  y are related by the following equation:  sin(x)+cos(y)=sqrt2 Also,  (dx)/(dt)=5. Find  (dy)/(dt) when  y=(pi)/(4) and  0 < x < (pi)/(2).

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Question Prompt: Question Prompt: Find (dy)/(dt) when y=(pi)/(4) and 0 < x < (pi)/(2) given that sin(x) + cos(y) = sqrt(2) and (dx)/(dt)=5.Differentiate Equation: Differentiate both sides of the equation sin(x) + cos(y) = sqrt(2) with respect to t. d/dt(sin(x)) + d/dt(cos(y)) = d/dt(sqrt(2)) Since sqrt(2) is a constant, its derivative with respect to t is 0.Apply Chain Rule: Apply the chain rule to differentiate sin(x) and cos(y) with respect to t. d/dt(sin(x)) = cos(x) * (dx/dt) and d/dt(cos(y)) = -sin(y) * (dy/dt) So, we have cos(x) * (dx/dt) - sin(y) * (dy/dt) = 0Substitute Given Values: Substitute the given values into the differentiated equation. Given (dx/dt) = 5 and y = (pi)/4, substitute these values to find (dy/dt). cos(x) * 5 - sin((pi)/4) * (dy/dt) = 0 Since sin((pi)/4) = sqrt(2)/2, the equation becomes cos(x) * 5 - (sqrt(2)/2) * (dy/dt) = 0Solve for (dy/dt): Solve for (dy/dt). Rearrange the equation to solve for (dy/dt): (dy/dt) = (cos(x) * 5) / (sqrt(2)/2) Since 0 < x < (pi)/2, cos(x) is positive. Given sin(x) + cos(y) = sqrt(2) and y = (pi)/4, cos(y) = sqrt(2)/2, so sin(x) = sqrt(2) - sqrt(2)/2 = sqrt(2)/2. Thus, cos(x) = sqrt(2)/2.Substitute cos(x): Substitute cos(x) = sqrt(2)/2 into the equation for (dy/dt). (dy/dt) = ((sqrt(2)/2) * 5) / (sqrt(2)/2) Simplifying gives (dy/dt) = 5.

The differentiable functions $x$ and $y$ are related by the following equation: $\newline$ $\sin (x)+\cos (y)=\sqrt{2}$ $\newline$ Also, $\frac{d x}{d t}=5$ . $\newline$ Find $\frac{d y}{d t}$ when $y=\frac{\pi}{4}$ and \( 0

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The differentiable functions $x$ and $y$ are related by the following equation: $\newline$ $\sin (x)+\cos (y)=\sqrt{2}$ $\newline$ Also, $\frac{d x}{d t}=5$ . $\newline$ Find $\frac{d y}{d t}$ when $y=\frac{\pi}{4}$ and \( 0