This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To find (dy)/(dx) for the equation y^3 x + 5 = 0, we will use implicit differentiation. Step 1: Differentiate both sides of the equation with respect to x. (d)/(dx)(y^3 x + 5) = (d)/(dx)(0) Step 2: Apply the product rule to y^3 x. The product rule states (d)/(dx)(uv) = u'v + uv'. Let u = y^3 and v = x. Then u' = (d)/(dx)(y^3) = 3y^2 (dy)/(dx) (using the chain rule). And v' = (d)/(dx)( x) = - x. The derivative of the constant 5 is 0. The derivative of 0 is 0. Substituting these into the equation: (3y^2 (dy)/(dx))( x) + (y^3)(- x) + 0 = 0 3y^2 x (dy)/(dx) - y^3 x = 0 Step 3: Isolate the term containing (dy)/(dx). 3y^2 x (dy)/(dx) = y^3 x Step 4: Solve for (dy)/(dx). (dy)/(dx) = (y^3 x)/(3y^2 x) Step 5: Simplify the expression. (dy)/(dx) = (y x)/(3 x) This can also be written using the identity x = ( x)/( x): (dy)/(dx) = (y)/(3) x The final answer is (dy)/(dx) = (y x)/(3 x). That's 2 down. 3 left today — send the next one.