Find dy/dx. x = ∛t, y = 4 - t

Solution:

We apply differentiation of parametric functions as x = f(t) and y = g(t)

Let us find dy/dx by finding dy/dt and dx/dt

dy/dx = dy/dt . dt/dx

x = ∛t= \(t^{\frac{1}{3}}\)

y  = 4 - t

\(\frac{\mathrm{d} y}{\mathrm{d} x} = \frac{\frac{\mathrm{d} y}{\mathrm{d} t}}{\frac{\mathrm{d} x}{\mathrm{d} t}}\)

\(\frac{\mathrm{d} y}{\mathrm{d} t} = 0 - \frac{\mathrm{d} t}{\mathrm{d} t}\) = -1

\(\frac{\mathrm{d}x }{\mathrm{d} t} = \frac{1}{3}t^{\frac{1}{3}-1}\)

\(\frac{\mathrm{d}x }{\mathrm{d} t} = \frac{1}{3}t^{\frac{2}{3}}\)

Now we know:

\(\frac{\mathrm{d} y}{\mathrm{d} x} = \frac{\frac{\mathrm{d} y}{\mathrm{d} t}}{\frac{\mathrm{d} x}{\mathrm{d} t}}\)

\(\frac{\mathrm{d} y}{\mathrm{d} x}   =\frac{-1}{\frac{1}{3}t^{\frac{2}{3}}}\)

dy/dx =-3/∛t²

Find dy/dx. x = ∛t, y = 4 - t

Summary:

dy/dx =-3/∛t² when   x = ∛t, and  y = 4 - t