A curve is represented by the equation 4y3 xy 5x 4 0 part a...

A curve is represented by the equation 4y^3 + xy - 5x + 4 = 0. Part A: Determine dy/dx (10 points) Part B: Evaluate dy/dx at the point (2, 1). (10 points) Part C: Determine d^2y/dx^2 (10 points) Part D: Evaluate d^2y/dx^2 at the point (2, 1). (10 points)

Transcript

00:01 All the student, given question is 4 y cube 4 y cube plus x multiplied by y minus 5 x plus 4 is equals to 0.

00:15 Now first question is we have to find dy divide by d x right and second question is you have to find dy divide by d x right and second question is you have to find dy divide by d x at the point 2 comma 1.

00:29 1 and the third question is we have to find out d square y divide by d x square next to fourth question is we have to find out d square y by d x square at the point 2 comma 1 now we have to differentiate given curve with respect to x what you will get differentiation of given curve right at 12 y square multiplied by y divide by d x right plus here we have to apply product rule right x multiplied by the y divide by d x plus y minus 5 is equal to zero right now if we take dy by d x common right what you will get x plus 12 y square right is equals to 5 minus y then dy by d x become right divide by d x is equal to 5 minus y divide by right x plus 12 y o square right this is our answer for first question right this is answer for first question now you have to find the d y by d x at the point 2 comma 1 right then then d .y, divide by d x at the point 2 comma 1 is equal to write 5 minus 1 divided by 2 plus 12 multiplied by 1 square is equal to write 4 divide by 14, right 7s are 2 2 2 2s, this is 2 divide by 7.

02:25 This is our answer for second question.

02:29 Next we have to find d square y by d x square.

02:33 Again, we have to differentiate d .y by dx with respect to x.

02:38 What will get that question? d square y divided by d x square is equal to.

02:46 Here we have to apply quotient rule.

02:50 Right.

02:51 Now this one can be written as 2 plus 12 y.

02:56 Sorry this is x right x plus 12 y x plus 12 y square then here x plus 12 y square then derivative of 5 minus y is minus d y divide by d x right minus 5 minus y multiplied by we have to differentiate this one what we will get 1 plus right 24 y multiplied by d .y by d .y by dx, right, dy divided by dx.

03:40 This is our required d square y by dx square...

Question

A curve is represented by the equation 4y^3 + xy - 5x + 4 = 0. Part A: Determine dy/dx (10 points) Part B: Evaluate dy/dx at the point (2, 1). (10 points) Part C: Determine d^2y/dx^2 (10 points) Part D: Evaluate d^2y/dx^2 at the point (2, 1). (10 points)

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