Find (dy)/(dx). (a) 4xy=sin(x+y)

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Find  (dy)/(dx). (a)  4xy=sin(x+y)

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Differentiate with respect to x: Differentiate both sides of the equation with respect to x. We have the equation 4xy = sin(x + y). To find (dy)/(dx), we need to differentiate both sides of the equation with respect to x using the product rule for the left side and the chain rule for the right side.Apply product rule: Apply the product rule to the left side. The product rule states that d(uv)/dx = u(dv/dx) + v(du/dx), where u = 4y and v = x. So, we differentiate 4y with respect to x, which is 0 since y is treated as a constant with respect to x, and x with respect to x, which is 1.Apply chain rule: Apply the chain rule to the right side. The chain rule states that d(f(g(x)))/dx = f'(g(x)) * g'(x). Here, f(u) = sin(u) and g(x) = x + y. The derivative of sin(u) with respect to u is cos(u), and the derivative of g(x) with respect to x is 1 + (dy)/(dx) since y is a function of x.Write down derivatives: Write down the derivatives. The derivative of the left side is 4(dy/dx)x + 4y(1), and the derivative of the right side is cos(x + y)(1 + (dy)/(dx)).Set equal to each other: Set the derivatives equal to each other. We have 4(dy/dx)x + 4y = cos(x + y)(1 + (dy)/(dx)).Solve for (dy)/(dx): Solve for (dy)/(dx). To solve for (dy)/(dx), we need to collect all terms containing (dy)/(dx) on one side and the remaining terms on the other side. This gives us 4(dy/dx)x - cos(x + y)(dy)/(dx) = cos(x + y) - 4y.Factor out (dy)/(dx): Factor out (dy)/(dx). Factoring (dy)/(dx) from the left side, we get (dy)/(dx)(4x - cos(x + y)) = cos(x + y) - 4y.Divide by (4x - cos(x + y)): Divide both sides by (4x - cos(x + y)). To isolate (dy)/(dx), we divide both sides by (4x - cos(x + y)), assuming that 4x - cos(x + y) is not equal to zero. This gives us (dy)/(dx) = (cos(x + y) - 4y) / (4x - cos(x + y)).

Find $\frac{dy}{dx}$ . $4xy=\sin(x+y)$

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Find $\frac{dy}{dx}$ . $4xy=\sin(x+y)$