Question

Find the derivative of the function y defined implicitly in terms of x. 8xy^3 + 9x^3y = 1 Step 1 of 5 Recall the rules of implicit differentiation, given that y is a function of x. We treat x and y independently while differentiating both sides of the equation. We compute the derivative with respect to x of expressions in x as we are accustomed. For example, we implicitly differentiate x^2 as follows. d/dx(x^2) = 2x Since y = y(x) is a function of x, any time we take the derivative with respect to x of an expression in y, we must use the chain rule. This creates the factor dy/dx. We will eventually solve for dy/dx which is the derivative of the implicitly defined function y. Step 2 of 5 In the given problem, we implicitly differentiate by taking the derivative with respect to x of both sides of the equation. d/dx(8xy^3 + 9x^3y) = d/dx(1) Evaluate the derivative with respect to x of the right-hand side of the equation, 1. d/dx(1) = (-27x^2y - 8y^3) / (24y^2x + 9x^3)

          Find the derivative of the function y defined implicitly in terms of x.
8xy^3 + 9x^3y = 1

Step 1 of 5
Recall the rules of implicit differentiation, given that y is a function of x.

We treat x and y independently while differentiating both sides of the equation. We compute the derivative with respect to x of expressions in x as we are accustomed. For example, we implicitly differentiate x^2 as follows.
d/dx(x^2) = 2x

Since y = y(x) is a function of x, any time we take the derivative with respect to x of an expression in y, we must use the chain rule. This creates the factor dy/dx. We will eventually solve for dy/dx which is the derivative of the implicitly defined function y.

Step 2 of 5
In the given problem, we implicitly differentiate by taking the derivative with respect to x of both sides of the equation.
d/dx(8xy^3 + 9x^3y) = d/dx(1)

Evaluate the derivative with respect to x of the right-hand side of the equation, 1.
d/dx(1) = (-27x^2y - 8y^3) / (24y^2x + 9x^3)

Find the derivative of the function y defined implicitly in terms of x.
8xy^3 + 9x^3y = 1

Step 1 of 5
Recall the rules of implicit differentiation, given that y is a function of x.

We treat x and y independently while differentiating both sides of the equation. We compute the derivative with respect to x of expressions in x as we are accustomed. For example, we implicitly differentiate x^2 as follows.
d/dx(x^2) = 2x

Since y = y(x) is a function of x, any time we take the derivative with respect to x of an expression in y, we must use the chain rule. This creates the factor dy/dx. We will eventually solve for dy/dx which is the derivative of the implicitly defined function y.

Step 2 of 5
In the given problem, we implicitly differentiate by taking the derivative with respect to x of both sides of the equation.
d/dx(8xy^3 + 9x^3y) = d/dx(1)

Evaluate the derivative with respect to x of the right-hand side of the equation, 1.
d/dx(1) = (-27x^2y - 8y^3) / (24y^2x + 9x^3)

Added by Christopher T.

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