Consider the function
$$z(x,y) = (3x-4)^{y^2+1}.$$
Determine $z'_x(2,3)$.
$$z(x,y) = (3x-4)^{y^2+1}.$$
Determine $z'_x(2,3)$.
Antwoord 1 correct
Correct
Antwoord 2 optie
$z'_x(2,3) = 6144\ln(2).$
Antwoord 2 correct
Fout
Antwoord 3 optie
$z'_x(2,3) = 5120.$
Antwoord 3 correct
Fout
Antwoord 4 optie
$z'_x(2,3) = 9375.$
Antwoord 4 correct
Fout
Antwoord 1 optie
None of the other options is correct.
Antwoord 1 feedback
Correct: We have to determine the partial derivative with respect to $x$. Hence, we can consider $y$, and also $y^2+1$, as a constant. Then the partial derivative of $z(x,y)$ with respect to $x$ is:
$$z'_x(x,y) = (y^2+1)(3x-4)^{y^2+1-1}\cdot 3 = 3(y^2+1)(3x-4)^{y^2}.$$
Finally, we plug in $(x,y)=(2,3)$:
$$z'_x(2,3) = 3(3^2+1)(3\cdot 2-4)^{3^2} = 3 \cdot 10 \cdot 2^9 = 15360.$$
Go on.
$$z'_x(x,y) = (y^2+1)(3x-4)^{y^2+1-1}\cdot 3 = 3(y^2+1)(3x-4)^{y^2}.$$
Finally, we plug in $(x,y)=(2,3)$:
$$z'_x(2,3) = 3(3^2+1)(3\cdot 2-4)^{3^2} = 3 \cdot 10 \cdot 2^9 = 15360.$$
Go on.
Antwoord 2 feedback
Wrong: With respect to which variable should $z(x,y)$ be differentiated?
See Partial derivatives, Example 1, and Example 3.
See Partial derivatives, Example 1, and Example 3.
Antwoord 3 feedback
Wrong: Do not forget to use the composite power rule.
See Extra explanation: special cases, Example 1, Example 2 and Example 3.
See Extra explanation: special cases, Example 1, Example 2 and Example 3.
Antwoord 4 feedback
Wrong: You probably plugged in the wrong values for $x$ and $y$.
Try again.
Try again.