Functions z = f(x, y), x = g(t) and y = h(t) are given.
(a) Use the Multivariable Chain Rule to compute \(\frac{dz}{dt}\)
(b) Evaluate \(\frac{dz}{dt}\) at the indicated t-value.
Calculate what’s given:
\[\begin{cases} z = 3x + 4y \begin{cases}f_x(x,y) = \frac{dz}{dx} = 3 \\ f_y(x,y) = \frac{dz}{dy} = 4 \end{cases} \\ x = t^2 => \begin{cases} \frac{dx}{dt} = -2t \end{cases} \\ y = 2t => \begin{cases} \frac{dy}{dt} = 2 \end{cases}\\ t = 1 \end{cases}\]
Concluding, \(\frac{dz}{dt} = 3(-2t) + 4(2) = -6t + 8\)
t <- 1
(dz_dt <- -6*t + 8)## [1] 2