given the equation x^7 + y^6 = 21, find dy/dx by implicit differentiation.

Explanation:

Step1: Differentiate both sides

Differentiate $x^{7}+y^{6}=21$ with respect to $x$. $\frac{d}{dx}(x^{7})+\frac{d}{dx}(y^{6})=\frac{d}{dx}(21)$

Step2: Apply power - rule and chain - rule

For $\frac{d}{dx}(x^{7})$, by power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$, we have $\frac{d}{dx}(x^{7}) = 7x^{6}$. For $\frac{d}{dx}(y^{6})$, by chain - rule $\frac{d}{dx}(u^{n})=nu^{n - 1}\frac{du}{dx}$ (where $u = y$), we get $6y^{5}\frac{dy}{dx}$. And $\frac{d}{dx}(21)=0$. So, $7x^{6}+6y^{5}\frac{dy}{dx}=0$.

Step3: Solve for $\frac{dy}{dx}$

Subtract $7x^{6}$ from both sides: $6y^{5}\frac{dy}{dx}=-7x^{6}$. Then divide both sides by $6y^{5}$ (assuming $y\neq0$): $\frac{dy}{dx}=-\frac{7x^{6}}{6y^{5}}$.

Answer:

$-\frac{7x^{6}}{6y^{5}}$