exercises 3.8 implicit differentiation\nscore: 12/20 answered: 6/10\nprogress saved done \nquestion 7\n0/2…

exercises 3.8 implicit differentiation\nscore: 12/20 answered: 6/10\nprogress saved done \nquestion 7\n0/2 pts 100 99 details\ngiven the equation below, find \\( \\frac { d y } { d x } \\).\n\\( - 11 x ^ { 7 } + 2 x ^ { 22 } y + y ^ { 9 } = - 8 \\)\nhint: you will need to use the product rule on the middle\nterm.\n\\( \\frac { d y } { d x } = \\)\nnow, find the equation of the tangent line to the curve at\n\\( ( 1,1 ) \\). write your answer in \\( m x + b \\) format\n\\( y = \\)\nquestion help: video message instructor\nsubmit question jump to answer

exercises 3.8 implicit differentiation\nscore: 12/20 answered: 6/10\nprogress saved done \nquestion 7\n0/2 pts 100 99 details\ngiven the equation below, find \\( \\frac { d y } { d x } \\).\n\\( - 11 x ^ { 7 } + 2 x ^ { 22 } y + y ^ { 9 } = - 8 \\)\nhint: you will need to use the product rule on the middle\nterm.\n\\( \\frac { d y } { d x } = \\)\nnow, find the equation of the tangent line to the curve at\n\\( ( 1,1 ) \\). write your answer in \\( m x + b \\) format\n\\( y = \\)\nquestion help: video message instructor\nsubmit question jump to answer

Answer

Explanation:

Step1: Differentiate each term with respect to (x)

  • For (-11x^{7}), using the power rule (\frac{d}{dx}(x^n)=nx^{n - 1}), we get (\frac{d}{dx}(-11x^{7})=-77x^{6}).
  • For (2x^{22}y), using the product rule ((uv)^\prime = u^\prime v+uv^\prime) where (u = 2x^{22}) and (v=y). Then (u^\prime=44x^{21}) and (v^\prime=\frac{dy}{dx}), so (\frac{d}{dx}(2x^{22}y)=44x^{21}y + 2x^{22}\frac{dy}{dx}).
  • For (y^{9}), using the chain rule (\frac{d}{dx}(y^n)=ny^{n - 1}\frac{dy}{dx}), we get (\frac{d}{dx}(y^{9})=9y^{8}\frac{dy}{dx}).
  • The derivative of the constant (-8) is (0).

So, (-77x^{6}+44x^{21}y + 2x^{22}\frac{dy}{dx}+9y^{8}\frac{dy}{dx}=0).

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

Group the terms with (\frac{dy}{dx}): ((2x^{22}+9y^{8})\frac{dy}{dx}=77x^{6}-44x^{21}y) Then (\frac{dy}{dx}=\frac{77x^{6}-44x^{21}y}{2x^{22}+9y^{8}}).

Step3: Find the slope (m) at the point ((1,1))

Substitute (x = 1) and (y = 1) into (\frac{dy}{dx}): (m=\frac{77(1)^{6}-44(1)^{21}(1)}{2(1)^{22}+9(1)^{8}}=\frac{77 - 44}{2+9}=\frac{33}{11}=3).

Step4: Find the equation of the tangent line

Using the point - slope form (y - y_1=m(x - x_1)) with (x_1 = 1), (y_1 = 1) and (m = 3). (y-1=3(x - 1)) Expand: (y-1=3x-3) Rearrange to (y=mx + b) form: (y=3x-2).

Answer:

(\frac{dy}{dx}=\frac{77x^{6}-44x^{21}y}{2x^{22}+9y^{8}}); (y = 3x-2)