Help with Implicit differentiation
y-y^3=x
To find the derivative of an implicitly defined equation like this, you need to use implicit differentiation. Here's how you can go about solving for dy/dx in the equation y - y^3 = x:
Step 1: Differentiate both sides of the equation with respect to x. Recall that when differentiating y with respect to x, you should treat y as a function of x and use the chain rule.
d/dx(y - y^3) = d/dx(x)
Step 2: On the left side, apply the chain rule to each term. For the first term, the derivative of y with respect to x is written as dy/dx, and for the second term, differentiate y^3 using the chain rule.
dy/dx - d/dx(y^3) = 1
Step 3: To differentiate y^3, apply the power rule by bringing down the exponent and reducing the exponent by 1. Multiply this by the derivative of the base, which is dy/dx, using the chain rule.
dy/dx - 3y^2(dy/dx) = 1
Step 4: Now, rearrange the equation to solve for dy/dx, which is the derivative of y with respect to x. Group the terms with dy/dx on one side and the constant terms on the other side.
dy/dx - 3y^2(dy/dx) = 1
dy/dx(1 - 3y^2) = 1
dy/dx = 1 / (1 - 3y^2)
So, the derivative of y with respect to x is given by dy/dx = 1 / (1 - 3y^2).
This is the result of applying implicit differentiation to the equation y - y^3 = x.