Suppose that u and v are differentiable functions of x and in particular that 𝑢𝑢(0) = 5, 𝑢𝑢′ (0) = −3, 𝑣𝑣(0) = −1, 𝑣𝑣′ (0) = 2&nbsp;Find the values of the following derivatives at x=0 𝑑𝑑 𝑑𝑑𝑑𝑑 (𝑢𝑢𝑣𝑣), 𝑑𝑑 𝑑𝑑𝑑𝑑 (7𝑣𝑣 − 2𝑢𝑢), 𝑑𝑑 𝑑𝑑𝑑𝑑 � 𝑢𝑢 𝑣𝑣

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Suppose that u and v are differentiable functions of x and in particular that 𝑢𝑢(0) = 5, 𝑢𝑢′ (0) = −3, 𝑣𝑣(0) = −1, 𝑣𝑣′ (0) = 2&amp;nbsp;Find the values of the following derivatives at x=0 𝑑𝑑 𝑑𝑑𝑑𝑑 (𝑢𝑢𝑣𝑣), 𝑑𝑑 𝑑𝑑𝑑𝑑 (7𝑣𝑣 − 2𝑢𝑢), 𝑑𝑑 𝑑𝑑𝑑𝑑 � 𝑢𝑢 𝑣𝑣

nick1337 · Accepted Answer

To solve the given problem, let&#039;s start by using the notation u′(x) to represent the derivative of function u(x) with respect to x. Similarly, v′(x) represents the derivative of function v(x) with respect to x.Given information:u(0)=5u′(0)=−3v(0)=−1v′(0)=2Now, we need to find expressions for u(x) and v(x) based on the given information. We&#039;ll use the fact that u′(x) and v′(x) are the derivatives of u(x) and v(x), respectively.Let&#039;s integrate u′(x) to find u(x). We have:u′(x)=−3Integrating both sides with respect to x, we get:∫u′(x)dx=∫−3dxIntegrating the left side gives us:u(x)=−3x+C1where C1 is the constant of integration.Since u(0)=5, we can substitute this into the equation:5=−3(0)+C1Simplifying, we find:C1=5Therefore, the expression for u(x) is:u(x)=−3x+5Next, let&#039;s integrate v′(x) to find v(x). We have:v′(x)=2Integrating both sides with respect to x, we get:∫v′(x)dx=∫2dxIntegrating the left side gives us:v(x)=2x+C2where C2 is the constant of integration.Since v(0)=−1, we can substitute this into the equation:−1=2(0)+C2Simplifying, we find:C2=−1Therefore, the expression for v(x) is:v(x)=2x−1

Suppose that u and v are differentiable functions

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