If x = y³ – y and dy/dt = 5, then what is dx/dt when y = 2?

Verified step by step guidance

1. First, recognize that you need to find the rate of change of x with respect to time, dx/dt, given the equation x = y³ - y and the rate of change of y with respect to time, dy/dt = 5.
2. To find dx/dt, use implicit differentiation with respect to time t on both sides of the equation x = y³ - y. This involves applying the chain rule.
3. Differentiate the left side of the equation with respect to t: d(x)/d(t) = dx/dt.
4. Differentiate the right side of the equation with respect to t: d(y³ - y)/d(t) = 3y²(dy/dt) - (dy/dt).
5. Substitute dy/dt = 5 and y = 2 into the differentiated equation to solve for dx/dt: dx/dt = 3(2)²(5) - (5).

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of a function when it is not explicitly solved for one variable in terms of another. In this problem, x is given in terms of y, and we need to differentiate both sides with respect to time t, applying the chain rule to account for dy/dt.

Chain Rule

The chain rule is a fundamental concept in calculus used to differentiate composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. Here, it helps relate dx/dt to dy/dt by differentiating x = y³ - y with respect to t.

Substitution

Substitution involves replacing variables with known values to simplify expressions or solve equations. In this problem, after finding the expression for dx/dt in terms of y and dy/dt, we substitute y = 2 and dy/dt = 5 to calculate the specific value of dx/dt at that point.