Question

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

Answer 1

Differentiate both sides of the equation.ddx(x3+y3)=ddx(18xy)ddx(x3+y3)=ddx(18xy)Differentiate the left side of the equation.Tap for fewer steps...By the Sum Rule, the derivative of x3+y3x3+y3 with respect to xx is ddx[x3]+ddx[y3]ddx[x3]+ddx[y3].ddx[x3]+ddx[y3]ddx[x3]+ddx[y3]Differentiate using the Power Rule which states that ddx[xn]ddx[xn] is nxn−1nxn-1 where n=3n=3.3x2+ddx[y3]3x2+ddx[y3]Evaluate ddx[y3]ddx[y3].Tap for more steps...3x2+3y2ddx[y]3x2+3y2ddx[y]Differentiate the right side of the equation.Tap for fewer steps...Since 1818 is constant with respect to xx, the derivative of 18xy18xy with respect to xx is 18ddx[xy]18ddx[xy].18ddx[xy]18ddx[xy]Differentiate using the Product Rule which states that ddx[f(x)g(x)]ddx[f(x)g(x)] is f(x)ddx[g(x)]+g(x)ddx[f(x)]f(x)ddx[g(x)]+g(x)ddx[f(x)] where f(x)=xf(x)=x and g(x)=yg(x)=y.18(xddx[y]+yddx[x])18(xddx[y]+yddx[x])Rewrite ddx[y]ddx[y] as ddx[y]ddx[y].18(xddx[y]+yddx[x])18(xddx[y]+yddx[x])Differentiate using the Power Rule which states that ddx[xn]ddx[xn] is nxn−1nxn-1 where n=1n=1.18(xddx[y]+y⋅1)18(xddx[y]+y⋅1)Multiply yy by 11 to get yy.18(xddx[y]+y)18(xddx[y]+y)Simplify.Tap for more steps...18xddx[y]+18y18xddx[y]+18yReform the equation by setting the left side equal to the right side.3x2+3y2y'=18xy'+18y3x2+3y2y′=18xy′+18ySince 18xy'18xy′ contains the variable to solve for, move it to the left side of the equation by subtracting 18xy'18xy′ from both sides.3x2+3y2y'−18xy'=18y3x2+3y2y′-18xy′=18ySince 3x23x2 does not contain the variable to solve for, move it to the right side of the equation by subtracting 3x23x2 from both sides.3y2y'−18xy'=−3x2+18y3y2y′-18xy′=-3x2+18yFactor 3y'3y′ out of 3y2y'−18xy'3y2y′-18xy′.Tap for fewer steps...Factor 3y'3y′ out of 3y2y'3y2y′.3y'(y2)−18xy'=−3x2+18y3y′(y2)-18xy′=-3x2+18yFactor 3y'3y′ out of −18xy'-18xy′.3y'(y2)+3y'(−6x)=−3x2+18y3y′(y2)+3y′(-6x)=-3x2+18yFactor 3y'3y′ out of 3y'y2+3y'(−6x)3y′y2+3y′(-6x).3y'(y2−6x)=−3x2+18y3y′(y2-6x)=-3x2+18yDivide each term by y2−6xy2-6x and simplify.Tap for fewer steps...Divide each term in 3y'(y2−6x)=−3x2+18y3y′(y2-6x)=-3x2+18y by y2−6xy2-6x.3y'(y2−6x)y2−6x=−3x2y2−6x+18yy2−6x3y′(y2-6x)y2-6x=-3x2y2-6x+18yy2-6xReduce the expression by cancelling the common factors.Tap for more steps...3y'=−3x2y2−6x+18yy2−6x3y′=-3x2y2-6x+18yy2-6xSimplify the right side of the equation.Tap for more steps...3y'=−3(x2−6y)y2−6x3y′=-3(x2-6y)y2-6xDivide each term by 33 and simplify.Tap for fewer steps...Divide each term in 3y'=−3(x2−6y)y2−6x3y′=-3(x2-6y)y2-6x by 33.3y'3=−3(x2−6y)y2−6x33y′3=-3(x2-6y)y2-6x3Reduce the expression by cancelling the common factors.Tap for more steps...y'=−3(x2−6y)y2−6x3y′=-3(x2-6y)y2-6x3Simplify the right side of the equation.Tap for more steps...y'=−x2−6yy2−6xy′=-x2-6yy2-6xReplace y'y′ with dydxdydx.dydx=−x2−6yy2−6x