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

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 $(3, - 1)$

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

Step4: Use the point - slope form to find the equation of the tangent line

The point - slope form of a line is $y - y_{1}=m(x - x_{1})$, where $(x_{1},y_{1})=(3,-1)$ and $m = 27$. Substitute these values: $y+1=27(x - 3)$. Expand to get $y+1=27x-81$, and then $y=27x-82$.

Answer:

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