exercises 3.8 implicit differentiation\nscore: 6/20 answered: 3/10\nprogress saved done\nquestion 4\n0/2 pts…

exercises 3.8 implicit differentiation\nscore: 6/20 answered: 3/10\nprogress saved done\nquestion 4\n0/2 pts 100 99 details\ntextbook videos +\nuse implicit differentiation to determine \\( \\frac{d y}{d x} \\) given the\nequation \\( x^{6}+y^{4}=\\cos y \\).\n\\( \\frac{d y}{d x}= \\)\nquestion help: message instructor\nsubmit question jump to answer

exercises 3.8 implicit differentiation\nscore: 6/20 answered: 3/10\nprogress saved done\nquestion 4\n0/2 pts 100 99 details\ntextbook videos +\nuse implicit differentiation to determine \\( \\frac{d y}{d x} \\) given the\nequation \\( x^{6}+y^{4}=\\cos y \\).\n\\( \\frac{d y}{d x}= \\)\nquestion help: message instructor\nsubmit question jump to answer

Answer

Explanation:

Step1: Differentiate both sides with respect to (x)

Differentiate (x^{6}+y^{4}=\cos y) term - by - term. Using the power rule (\frac{d}{dx}(x^{n})=nx^{n - 1}), for the left - hand side: (\frac{d}{dx}(x^{6})+\frac{d}{dx}(y^{4})). We know that (\frac{d}{dx}(x^{6}) = 6x^{5}), and for (\frac{d}{dx}(y^{4})), by the chain rule (\frac{d}{dx}(y^{4})=4y^{3}\frac{dy}{dx}). For the right - hand side, using the chain rule (\frac{d}{dx}(\cos y)=-\sin y\frac{dy}{dx}). So we have (6x^{5}+4y^{3}\frac{dy}{dx}=-\sin y\frac{dy}{dx}).

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

First, move all terms with (\frac{dy}{dx}) to one side: (4y^{3}\frac{dy}{dx}+\sin y\frac{dy}{dx}=-6x^{5}). Factor out (\frac{dy}{dx}): (\frac{dy}{dx}(4y^{3}+\sin y)=-6x^{5}). Then, divide both sides by ((4y^{3}+\sin y)) to get (\frac{dy}{dx}=\frac{-6x^{5}}{4y^{3}+\sin y}).

Answer:

(\frac{dy}{dx}=\frac{-6x^{5}}{4y^{3}+\sin y})