Use implicit differentiation to find \( \frac{d y}{d x} \) \[ x^{3}+\sin y=x^{2} y^{5} \]

We start with the given equation:   x³ + sin(y) = x²·y⁵ Differentiate both sides with respect to x. Remember that y is a function of x, so when differentiating terms involving y, we must use the chain rule. Step 1: Differentiate the left-hand side:   d/dx [x³] = 3x²   d/dx [sin(y)] = cos(y)·(dy/dx) So, the derivative of the left-hand side is:   3x² + cos(y)·(dy/dx) Step 2: Differentiate the right-hand side (using the product rule):   x²·y⁵ is a product of x² and y⁵.   Let u = x² (so u' = 2x) and v = y⁵ (so v' = 5y⁴·(dy/dx)). Using the product rule:   d/dx [x²·y⁵] = u'·v + u·v' = 2x·y⁵ + x²·5y⁴·(dy/dx) Step 3: Write the differentiated equation:   3x² + cos(y)·(dy/dx) = 2x·y⁵ + 5x²·y⁴·(dy/dx) Step 4: Solve for dy/dx. First, collect all terms involving dy/dx on one side:   cos(y)·(dy/dx) - 5x²·y⁴·(dy/dx) = 2x·y⁵ - 3x² Factor out dy/dx:   dy/dx · [cos(y) - 5x²·y⁴] = 2x·y⁵ - 3x² Finally, solve for dy/dx by dividing both sides by [cos(y) - 5x²·y⁴]:   dy/dx = (2x·y⁵ - 3x²) / (cos(y) - 5x²·y⁴) Thus, the derivative dy/dx is:   dy/dx = (2x·y⁵ - 3x²)⁄(cos(y) - 5x²·y⁴)