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

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

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Given Equation and Derivative: We are given the equation sin(x) + cos(y) = √2 and the derivative dx/dt = 5. We need to find dy/dt when y = π/4 and x = 0. To do this, we will differentiate both sides of the given equation with respect to t using the chain rule.Differentiating sin(x): Differentiating sin(x) with respect to t gives us cos(x) * (dx/dt) because the derivative of sin(x) with respect to x is cos(x) and then we multiply by dx/dt due to the chain rule.Differentiating cos(y): Differentiating cos(y) with respect to t gives us -sin(y) * (dy/dt) because the derivative of cos(y) with respect to y is -sin(y) and then we multiply by dy/dt due to the chain rule.Differentiating constant: Now we differentiate √2 with respect to t, which is a constant, so its derivative is 0.Derivative Equation: Putting it all together, we get the following equation from the derivatives: cos(x) * (dx/dt) - sin(y) * (dy/dt) = 0Substitute Values: We substitute the given values into the differentiated equation: x = 0, y = π/4, and dx/dt = 5. This gives us: cos(0) * 5 - sin(π/4) * (dy/dt) = 0Solve for dy/dt: We know that cos(0) = 1 and sin(π/4) = √2/2. Substituting these values into the equation gives us: 1 * 5 - (√2/2) * (dy/dt) = 0Rearrange Equation: Solving for dy/dt, we get: 5 - (√2/2) * (dy/dt) = 0Isolate dy/dt: Rearrange the equation to solve for dy/dt: (√2/2) * (dy/dt) = 5Simplify Equation: Divide both sides by √2/2 to isolate dy/dt: dy/dt = 5 / (√2/2)Rationalize Denominator: Simplify the right side of the equation: dy/dt = 5 * (2/√2)Final Simplification: Simplify further by multiplying 5 by 2/√2: dy/dt = 10/√2To rationalize the denominator, multiply the numerator and the denominator by √2: dy/dt = (10/√2) * (√2/√2)Simplify the expression: dy/dt = 10√2/2Finally, we can simplify 10√2/2 to 5√2: dy/dt = 5√2

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

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