use implicit differentiation of the equation below to determine the slope of the g\n\nxy^{4}=-49;x =…

Explanation:

Step1: Differentiate both sides

Differentiate (xy^{4}=-49) with respect to (x) using the product rule ((uv)^\prime = u^\prime v+uv^\prime) (where (u = x), (v = y^{4})). The derivative of (x) with respect to (x) is (1), and for (y^{4}) with respect to (x), we use the chain rule (\frac{d}{dx}(y^{4})=4y^{3}\frac{dy}{dx}). So, (\frac{d}{dx}(xy^{4})=\frac{d}{dx}(-49)). (y^{4}+x\cdot4y^{3}\frac{dy}{dx}=0).

Step2: Solve for (\frac{dy}{dx})

Rearrange the equation (y^{4}+4xy^{3}\frac{dy}{dx}=0) for (\frac{dy}{dx}). First, move (y^{4}) to the other side: (4xy^{3}\frac{dy}{dx}=-y^{4}). Then, divide both sides by (4xy^{3}) (assuming (x\neq0) and (y\neq0)): (\frac{dy}{dx}=-\frac{y}{4x}).

Step3: Substitute (x =-\frac{1}{49}) and (y = - 7)

Substitute (x =-\frac{1}{49}) and (y=-7) into (\frac{dy}{dx}=-\frac{y}{4x}). (\frac{dy}{dx}=-\frac{-7}{4\times(-\frac{1}{49})}). (\frac{dy}{dx}=-\frac{7}{\frac{4}{49}}). Using the rule (\frac{a}{b/a}=a\times\frac{a}{b}), we have (\frac{dy}{dx}=-\frac{7\times49}{4}=-\frac{343}{4}).

Answer:

(-\frac{343}{4})