Problem 2 - Essential Calculus

Question

Find \(d y / d x\). $$ x=\sqrt[3]{t}, y=4-t $$

Step-by-Step Solution

Answer

\(dy/dx = - 3t^{2/3}\)

Step 1: Differentiate \(x\) and \(y\) with respect to \(t\)

Starting with \(x = t^{1/3}\) and \(y = 4 - t\), differentiate each equation with respect to \(t\) to get \(dx/dt = (1/3) t^{-2/3}\) and \(dy/dt = -1\).

Step 2: Use the chain rule to calculate \(dy/dx\)

Now, use the chain rule to find \(dy/dx = dy/dt / dx/dt\). Substitute the calculated values of \(dx/dt\) and \(dy/dt\) into this relation, and simplify to get \(dy/dx = - (1 / (1/3) t^{-2/3}) = - 3t^{2/3}\).