Question

use implicit differentiation to find $z$ and then evaluate $z$ at $(-7,6)$. $xz + 42 = 0$ $z=square$ $z\big|_{(-7,6)}=square$ (simplify your answer.)

Explanation:

Step1: Differentiate both sides

Differentiate $xz + 42=0$ with respect to $x$. Using the product - rule $(uv)^\prime = u^\prime v+uv^\prime$ where $u = x$ and $v = z$. The derivative of $x$ with respect to $x$ is $1$, and the derivative of $z$ with respect to $x$ is $z^\prime$. The derivative of a constant ($42$) is $0$. So, $(x)^\prime z+x(z)^\prime+(42)^\prime = 0$, which gives $z + xz^\prime+0 = 0$.

Step2: Solve for $z^\prime$

Isolate $z^\prime$ in the equation $z + xz^\prime=0$. Subtract $z$ from both sides: $xz^\prime=-z$. Then divide both sides by $x$ (assuming $x
eq0$) to get $z^\prime=-\frac{z}{x}$.

Step3: Evaluate $z^\prime$ at $(-7,6)$

Substitute $x=-7$ and $z = 6$ into $z^\prime$. So, $z^\prime|_{(-7,6)}=-\frac{6}{-7}=\frac{6}{7}$.

Answer:

$z^\prime=-\frac{z}{x}$
$z^\prime|_{(-7,6)}=\frac{6}{7}$