Question

carry out the following steps for the given curve. a. use implicit differentiation to find $\frac{dy}{dx}$. $\frac{dy}{dx}=-\frac{x^{3}}{y^{3}}$ b. find the slope of the curve at the given point. the slope of $x^{4}+y^{4}=272$ at $(4, - 2)$ is $square$. (simplify your answer.)

Explanation:

Step1: Recall implicit - differentiation result

We are given $\frac{dy}{dx}=-\frac{x^{3}}{y^{3}}$

Step2: Substitute the point values

We need to find the slope at the point $(4, - 2)$. Substitute $x = 4$ and $y=-2$ into $\frac{dy}{dx}$.
$\frac{dy}{dx}=-\frac{4^{3}}{(-2)^{3}}$

Step3: Calculate the power values

$4^{3}=4\times4\times4 = 64$ and $(-2)^{3}=(-2)\times(-2)\times(-2)=-8$

Step4: Perform the division

$\frac{dy}{dx}=-\frac{64}{-8}=8$

Answer:

$8$