exercises 3.8 implicit differentiation\nscore: 0/20 answered: 0/10\nprogress saved done\nquestion 1\n0/2 pts…

exercises 3.8 implicit differentiation\nscore: 0/20 answered: 0/10\nprogress saved done\nquestion 1\n0/2 pts 100 99 details\ntextbook videos +\nuse implicit differentiation to determine \\( \\frac{d y}{d x} \\) given the\nequation \\( x^{6}+y^{5}=-2 \\).\n\\( \\frac{d y}{d x}= \\)\nquestion help: video message instructor\nsubmit question jump to answer

exercises 3.8 implicit differentiation\nscore: 0/20 answered: 0/10\nprogress saved done\nquestion 1\n0/2 pts 100 99 details\ntextbook videos +\nuse implicit differentiation to determine \\( \\frac{d y}{d x} \\) given the\nequation \\( x^{6}+y^{5}=-2 \\).\n\\( \\frac{d y}{d x}= \\)\nquestion help: video message instructor\nsubmit question jump to answer

Answer

Explanation:

Step1: Differentiate both sides

Differentiate (x^{6}+y^{5}=-2) with respect to (x). Using the power rule (\frac{d}{dx}(x^{n}) = nx^{n - 1}) and the chain - rule (\frac{d}{dx}(y^{n})=ny^{n - 1}\frac{dy}{dx}). For the left - hand side: (\frac{d}{dx}(x^{6}+y^{5})=\frac{d}{dx}(x^{6})+\frac{d}{dx}(y^{5})). (\frac{d}{dx}(x^{6}) = 6x^{5}), and (\frac{d}{dx}(y^{5})=5y^{4}\frac{dy}{dx}). The right - hand side: (\frac{d}{dx}(-2)=0). So, (6x^{5}+5y^{4}\frac{dy}{dx}=0).

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

Subtract (6x^{5}) from both sides: (5y^{4}\frac{dy}{dx}=-6x^{5}). Then divide both sides by (5y^{4}) ((y\neq0)).

Answer:

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