find the slope of the curve at the given point.\n$8y^{8}+5x^{7}=3y + 10x$ at $(1,1)$\nthe slope of the curve…

Explanation:

Step1: Differentiate both sides with respect to (x)

Using implicit differentiation: For (8y^{8}), by the chain - rule (\frac{d}{dx}(8y^{8})=8\times8y^{7}\frac{dy}{dx}=64y^{7}\frac{dy}{dx}) For (5x^{7}), (\frac{d}{dx}(5x^{7}) = 35x^{6}) For (3y), by the chain - rule (\frac{d}{dx}(3y)=3\frac{dy}{dx}) For (10x), (\frac{d}{dx}(10x)=10)

The derivative of the equation (8y^{8}+5x^{7}=3y + 10x) is: (64y^{7}\frac{dy}{dx}+35x^{6}=3\frac{dy}{dx}+10)

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

Rearrange the terms to isolate (\frac{dy}{dx}): (64y^{7}\frac{dy}{dx}-3\frac{dy}{dx}=10 - 35x^{6}) Factor out (\frac{dy}{dx}): (\frac{dy}{dx}(64y^{7}-3)=10 - 35x^{6}) So, (\frac{dy}{dx}=\frac{10 - 35x^{6}}{64y^{7}-3})

Step3: Substitute (x = 1) and (y = 1)

When (x = 1) and (y = 1): (\frac{dy}{dx}=\frac{10-35\times1^{6}}{64\times1^{7}-3}=\frac{10 - 35}{64 - 3}=\frac{-25}{61})

Answer:

(\frac{-25}{61})