Question

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, $\frac{d}{dx}y^{8}=(square)(y)$.

Explanation:

Step1: Apply the power - rule and chain - rule

Let \(u = y\), and we want to find the derivative of \(u^{8}\) with respect to \(x\). By the power - rule \(\frac{d}{du}(u^{n})=nu^{n - 1}\) and the chain - rule \(\frac{d}{dx}(u^{n})=\frac{d}{du}(u^{n})\cdot\frac{du}{dx}\).
For \(n = 8\) and \(u = y\), we have \(\frac{d}{dx}(y^{8})=\frac{d}{dy}(y^{8})\cdot\frac{dy}{dx}\).
Since \(\frac{d}{dy}(y^{8}) = 8y^{7}\), then \(\frac{d}{dx}(y^{8})=(8y^{7})(y')\).

Answer:

\(8y^{7}\)