Find dy/dx x^3+y^3=18xy

Differentiate both sides of the equation. d dx (x3 + y3) = d dx (18xy) d dx (x3 + y3) = d dx (18xy) Differentiate the left side of the equation. Tap for fewer steps ... By the Sum Rule, the derivative of x3 + y3 x3 + y3 with respect to xx is d dx [ x3 ] + d dx [ y3 ] d dx [ x3 ] + d dx [ y3 ]. d dx [ x3 ] + d dx [ y3 ] d dx [ x3 ] + d dx [ y3 ] Differentiate using the Power Rule which states that d dx [ xn ] d dx [ xn ] is n x n-1 n x n-1 where n=3 n=3. 3 x2 + d dx [ y3 ] 3 x2 + d dx [ y3 ] Evaluate d dx [ y3 ] d dx [ y3 ]. Tap for more steps ... 3 x2 + 3 y2 d dx [y] 3 x2 + 3 y2 d dx [y] Differentiate the right side of the equation. Tap for fewer steps ... Since 1818 is constant with respect to xx, the derivative of 18xy 18xy with respect to xx is 18 d dx [xy] 18 d dx [xy]. 18 d dx [xy] 18 d dx [xy] Differentiate using the Product Rule which states that d dx [f (x) g (x) ] d dx [f (x) g (x) ] is f (x) d dx [g (x) ] + g (x) d dx [f (x) ] f (x) d dx [g (x) ] + g (x) d dx [f (x) ] where f (x) = x f (x) = x and g (x) = y g (x) = y. 18 (x d dx [y] + y d dx [x]) 18 (x d dx [y] + y d dx [x]) Rewrite d dx [y] d dx [y] as d dx [y] d dx [y]. 18 (x d dx [y] + y d dx [x]) 18 (x d dx [y] + y d dx [x]) Differentiate using the Power Rule which states that d dx [ xn ] d dx [ xn ] is n x n-1 n x n-1 where n=1 n=1. 18 (x d dx [y] + y⋅1) 18 (x d dx [y] + y⋅1) Multiply yy by 11 to get yy. 18 (x d dx [y] + y) 18 (x d dx [y] + y) Simplify. Tap for more steps ... 18x d dx [y] + 18y 18x d dx [y] + 18y Reform the equation by setting the left side equal to the right side. 3 x2 + 3 y2 y'=18xy'+18y 3 x2 + 3 y2 y′=18xy′+18y Since 18xy' 18xy′ contains the variable to solve for, move it to the left side of the equation by subtracting 18xy' 18xy′ from both sides. 3 x2 + 3 y2 y'-18xy'=18y 3 x2 + 3 y2 y′-18xy′=18y Since 3 x2 3 x2 does not contain the variable to solve for, move it to the right side of the equation by subtracting 3 x2 3 x2 from both sides. 3 y2 y'-18xy'=-3 x2 + 18y 3 y2 y′-18xy′=-3 x2 + 18y Factor 3y' 3y′ out of 3 y2 y'-18xy' 3 y2 y′-18xy′. Tap for fewer steps ... Factor 3y' 3y′ out of 3 y2 y' 3 y2 y′. 3y' (y2) - 18xy'=-3 x2 + 18y 3y′ (y2) - 18xy′=-3 x2 + 18y Factor 3y' 3y′ out of - 18xy' - 18xy′. 3y' (y2) + 3y' (-6x) = - 3 x2 + 18y 3y′ (y2) + 3y′ (-6x) = - 3 x2 + 18y Factor 3y' 3y′ out of 3y' y2 + 3y' (-6x) 3y′ y2 + 3y′ (-6x). 3y' (y2 - 6x) = - 3 x2 + 18y 3y′ (y2 - 6x) = - 3 x2 + 18y Divide each term by y2 - 6x y2 - 6x and simplify. Tap for fewer steps ... Divide each term in 3y' (y2 - 6x) = - 3 x2 + 18y 3y′ (y2 - 6x) = - 3 x2 + 18y by y2 - 6x y2 - 6x. 3y' (y2 - 6x) y2 - 6x = - 3 x2 y2 - 6x + 18y y2 - 6x 3y′ (y2 - 6x) y2 - 6x = - 3 x2 y2 - 6x + 18y y2 - 6x Reduce the expression by cancelling the common factors. Tap for more steps ... 3y' = - 3 x2 y2 - 6x + 18y y2 - 6x 3y′ = - 3 x2 y2 - 6x + 18y y2 - 6x Simplify the right side of the equation. Tap for more steps ... 3y' = - 3 (x2 - 6y) y2 - 6x 3y′ = - 3 (x2 - 6y) y2 - 6x Divide each term by 33 and simplify. Tap for fewer steps ... Divide each term in 3y' = - 3 (x2 - 6y) y2 - 6x 3y′ = - 3 (x2 - 6y) y2 - 6x by 33. 3y' 3 = - 3 (x2 - 6y) y2 - 6x 3 3y′ 3 = - 3 (x2 - 6y) y2 - 6x 3 Reduce the expression by cancelling the common factors. Tap for more steps ... y' = - 3 (x2 - 6y) y2 - 6x 3 y′ = - 3 (x2 - 6y) y2 - 6x 3 Simplify the right side of the equation. Tap for more steps ... y' = - x2 - 6y y2 - 6x y′ = - x2 - 6y y2 - 6x Replace y' y′ with dy dx dy dx. dy dx = - x2 - 6y y2 - 6x