question 13 of 15, step 1 of 1 use implicit differentiation to determine dy/dx for the equation x + y^10 =…

Explanation:

Step1: Differentiate both sides

Differentiate $x + y^{10}=6$ with respect to $x$. The derivative of $x$ with respect to $x$ is $1$, and for $y^{10}$ we use the chain - rule. The derivative of a constant $6$ with respect to $x$ is $0$. So, $\frac{d}{dx}(x)+\frac{d}{dx}(y^{10})=\frac{d}{dx}(6)$. $1 + 10y^{9}\frac{dy}{dx}=0$

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

Isolate $\frac{dy}{dx}$ in the equation $1 + 10y^{9}\frac{dy}{dx}=0$. First, subtract $1$ from both sides: $10y^{9}\frac{dy}{dx}=- 1$. Then divide both sides by $10y^{9}$ (assuming $y\neq0$). $\frac{dy}{dx}=-\frac{1}{10y^{9}}$

Answer:

$-\frac{1}{10y^{9}}$