Implicit differemation 3) \( \frac{x^{2}}{y}=\cos \left(x^{2}\right) \)

We start with the equation   (x²)/y = cos(x²). Our goal is to differentiate both sides with respect to x, remembering that y is a function of x. Step 1. Differentiate the left-hand side (LHS): The left-hand side is (x²)/y, a quotient of x² and y. Using the quotient rule,   d/dx[(x²)/y] = [y · d/dx(x²) – x² · d/dx(y)] / y². Since d/dx(x²) = 2x and d/dx(y) = y' (using implicit differentiation),   d/dx[(x²)/y] = [y · 2x – x² · y'] / y². Step 2. Differentiate the right-hand side (RHS): The right-hand side is cos(x²). Using the chain rule,   d/dx[cos(x²)] = –sin(x²) · d/dx(x²) = – sin(x²) · 2x           = – 2x sin(x²). Step 3. Set the derivatives equal to each other:   (2x y – x² y')/y² = –2x sin(x²). Step 4. Solve for y': Multiply both sides by y² to eliminate the denominator:   2x y – x² y' = –2x y² sin(x²). Now, isolate the term containing y':   – x² y' = –2x y² sin(x²) – 2x y. Multiply both sides by –1:   x² y' = 2x y² sin(x²) + 2x y. Notice that 2x y is common in the right-hand side:   x² y' = 2x y (y sin(x²) + 1). Finally, solve for y' by dividing both sides by x² (assuming x ≠ 0):   y' = [2x y (y sin(x²) + 1)]/x² = 2y (y sin(x²) + 1)/x. Thus, the derivative of y with respect to x is   y' = (2y (y sin(x²) + 1))/x. This is the required result obtained by implicit differentiation.