Question

step 1
when finding $\frac{dy}{dx}=y$ by implicit differentiation, we consider $y$ to be a function of $x$. this means that whenever a derivative is calculated for an expression that includes the variable $y$, the chain rule requires that we multiply the derivative by $y$. for example, $\frac{d}{dx}y^{2}=(2y)(y)$.
similarly,
$\frac{d}{dx}y^{8}=(8y^{7})(y)$.
step 2
to find $\frac{dy}{dx}=y$ for $x^{3}+y^{8}=6$, we differentiate both sides of the equation with respect to $x$.
on the right - hand side, we have the following.
$\frac{d}{dx}6=square$
on the left - hand side, we have $\frac{d}{dx}x^{3}+y^{8}$. we already know that $\frac{d}{dx}y^{8}=8y^{7}y$. for the term $x^{3}$, we have the following.
$\frac{d}{dx}x^{3}=square$

Explanation:

Step1: Recall derivative of a constant

The derivative of a constant $c$ with respect to $x$ is 0. So, $\frac{d}{dx}[6]=0$.

Step2: Recall power - rule for differentiation

The power - rule states that $\frac{d}{dx}[x^n]=nx^{n - 1}$. For $n = 3$, we have $\frac{d}{dx}[x^3]=3x^{2}$.

Answer:

$\frac{d}{dx}[6]=0$, $\frac{d}{dx}[x^3]=3x^{2}$