Consider the function $Z(t) = z(x(t),y(t))$, where
$$z(x,y) = e^{2x + 3y}, \quad x(t) = 5t^2 \quad \text{and} \quad y(t) = \ln(2t).$$
Determine $Z'(t)$.
$$z(x,y) = e^{2x + 3y}, \quad x(t) = 5t^2 \quad \text{and} \quad y(t) = \ln(2t).$$
Determine $Z'(t)$.
Antwoord 1 correct
Correct
Antwoord 2 optie
$Z'(t) = e^{10t^2 + 3\ln(2t)}(10t + t^{-1}).$
Antwoord 2 correct
Fout
Antwoord 3 optie
$Z'(t) = 5e^{10t^2 + 3\ln(2t)}.$
Antwoord 3 correct
Fout
Antwoord 4 optie
$Z'(t) = e^{10t^2 + 3\ln(2t)}(2+ 3t^{-1}).$
Antwoord 4 correct
Fout
Antwoord 1 optie
$Z'(t) = e^{10t^2 + 3\ln(2t)}(20t + 3t^{-1}).$
Antwoord 1 feedback
Correct: The partial derivatives of $z(x,y)$ at $(x(t),y(t))$, the derivative of $x(t)$ and the derivative of $y(t)$ are:
$$
\begin{align*}
z'_x(x,y) &= e^{2x+3y}\cdot 2 = 2e^{2x+3y}\\
z'_x(x(t),y(t)) &= 2e^{2\cdot5t^2+3\cdot\ln(2t)} = 2e^{10t^2 + 3\ln(2t)}\\[2mm]
z'_y(x,y) &= e^{2x+3y}\cdot 3 = 3e^{2x+3y}\\
z'_y(x(t),y(t)) &= 3e^{2\cdot5t^2+3\cdot\ln(2t)} = 3e^{10t^2 + 3\ln(2t)}\\[2mm]
x'(t) &= 5 \cdot 2t = 10t\\[1mm]
y'(t) &= \tfrac{1}{2t} \cdot 2 = \tfrac{2}{2t} = \tfrac{1}{t}.
\end{align*}
$$
According to the Chain rule (case 1):
$$
Z'(t) = 2e^{10t^2 + 3\ln(2t)} \cdot 10t + 3e^{10t^2 + 3\ln(2t)} \cdot \tfrac{1}{t} = e^{10t^2 + 3\ln(2t)}(20t + 3t^{-1}).
$$
Go on.
$$
\begin{align*}
z'_x(x,y) &= e^{2x+3y}\cdot 2 = 2e^{2x+3y}\\
z'_x(x(t),y(t)) &= 2e^{2\cdot5t^2+3\cdot\ln(2t)} = 2e^{10t^2 + 3\ln(2t)}\\[2mm]
z'_y(x,y) &= e^{2x+3y}\cdot 3 = 3e^{2x+3y}\\
z'_y(x(t),y(t)) &= 3e^{2\cdot5t^2+3\cdot\ln(2t)} = 3e^{10t^2 + 3\ln(2t)}\\[2mm]
x'(t) &= 5 \cdot 2t = 10t\\[1mm]
y'(t) &= \tfrac{1}{2t} \cdot 2 = \tfrac{2}{2t} = \tfrac{1}{t}.
\end{align*}
$$
According to the Chain rule (case 1):
$$
Z'(t) = 2e^{10t^2 + 3\ln(2t)} \cdot 10t + 3e^{10t^2 + 3\ln(2t)} \cdot \tfrac{1}{t} = e^{10t^2 + 3\ln(2t)}(20t + 3t^{-1}).
$$
Go on.
Antwoord 2 feedback
Antwoord 3 feedback
Wrong: Do not forget to apply the chain rule: special case 1.
See Chain rule (case 1), Example 1 and Example 2.
See Chain rule (case 1), Example 1 and Example 2.
Antwoord 4 feedback
Wrong: We are dealing with special case 1 of the chain rule, not special case 2.
See Chain rule (case 1), Example 1 and Example 2.
See Chain rule (case 1), Example 1 and Example 2.