Question

suppose that $x^{4}+y^{4}=82$. (1) use the method of implicit differentiation to find $\frac{dy}{dx}$. $\frac{dy}{dx}=$ (2) find the equation of the tangent line at the point $(x,y)=(-3, - 1)$. the equation is $y=$

Explanation:

Step1: Differentiate both sides

Differentiate $x^{4}+y^{4}=82$ with respect to $x$. The derivative of $x^{4}$ with respect to $x$ is $4x^{3}$ by the power - rule. For $y^{4}$, using the chain - rule, we get $4y^{3}\frac{dy}{dx}$. The derivative of the constant 82 is 0. So, $4x^{3}+4y^{3}\frac{dy}{dx}=0$.

Step2: Solve for $\frac{dy}{dx}$

Subtract $4x^{3}$ from both sides: $4y^{3}\frac{dy}{dx}=-4x^{3}$. Then divide both sides by $4y^{3}$ (assuming $y
eq0$) to get $\frac{dy}{dx}=-\frac{x^{3}}{y^{3}}$.

Step3: Find the slope of the tangent line at the point $(-3,-1)$

Substitute $x = - 3$ and $y=-1$ into $\frac{dy}{dx}$. So, $m=\frac{dy}{dx}\big|_{x = - 3,y=-1}=-\frac{(-3)^{3}}{(-1)^{3}}=-27$.

Step4: Find the equation of the tangent line

Use the point - slope form of a line $y - y_{1}=m(x - x_{1})$, where $(x_{1},y_{1})=(-3,-1)$ and $m=-27$.
$y+1=-27(x + 3)$.
Expand to get $y+1=-27x-81$.
Solve for $y$: $y=-27x-82$.

Answer:

(1) $-\frac{x^{3}}{y^{3}}$
(2) $y=-27x - 82$