when finding $\frac{dy}{dx}=y$ by implicit differentiation, we consider y to be a function of x. this means…

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

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

Answer

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})