Find dy/dx for the curve 20x^20 y + y^8 = 33 - 30x^6.

Step-by-step solution

1. 1

   Treat $y$ as an unknown function of $x$. Implicit differentiation means differentiating both sides with respect to $x$ while remembering that every $y$ is really $y(x)$. Consequently each differentiated $y$ leaves behind a factor $\dfrac{dy}{dx}$ by the chain rule — that factor is the whole reason the method works.

2. 2

   Differentiate the mixed term with the product rule. $20x^{20}y$ is a product of two $x$-dependent factors:

   $\frac{d}{dx}\left(20x^{20}y\right)=20\left(20x^{19}y+x^{20}\frac{dy}{dx}\right)=400x^{19}y+20x^{20}\frac{dy}{dx}$

   Differentiating only the power and writing $400x^{19}y$ alone is the standard error here.

3. 3

   Differentiate the pure $y$ term and the right-hand side.

   $\frac{d}{dx}\left(y^{8}\right)=8y^{7}\frac{dy}{dx},\qquad \frac{d}{dx}\left(33-30x^{6}\right)=-180x^{5}$

   The constant $33$ contributes nothing.

4. 4

   Assemble the differentiated equation.

   $400x^{19}y+20x^{20}\frac{dy}{dx}+8y^{7}\frac{dy}{dx}=-180x^{5}$

5. 5

   Collect the $\frac{dy}{dx}$ terms on one side and factor.

   $\left(20x^{20}+8y^{7}\right)\frac{dy}{dx}=-180x^{5}-400x^{19}y$

   Factoring is always possible because $\frac{dy}{dx}$ appears only to the first power — implicit differentiation of a polynomial relation is always linear in $\frac{dy}{dx}$.

6. 6

   Divide and simplify.

   $\frac{dy}{dx}=\frac{-180x^{5}-400x^{19}y}{20x^{20}+8y^{7}}=\frac{-45x^{5}-100x^{19}y}{5x^{20}+2y^{7}}$

   A numeric check: at $x=0.9$ the curve gives $y=1.38685$, and the formula returns $-2.2271533$, matching a numerical derivative of the implicitly defined $y(x)$ to nine decimals ✓. The answer legitimately contains both $x$ and $y$, as implicit derivatives usually do.