Implicit differentiation: ycosx=x^2+y^2
derivative of both sides:
=(()+(cosx)(dy/dx)=2x+2y(dy/dx)
=(-ysinx+cosx(dy/dx)=2x+2y(dy/dx) =cosx(dy/dx) - 2y(dy/dx) = ysinx+2x
=(cosx-2y)dy/dx = ysinx+2x
=dy/dx = (ysinx+2x)/(cosx-2y)

Step-by-step explanation:

Given is an implicit function of y in x.

The steps shown were the differentiation on both the sides

Left side done using product rule as well as chain rule and right side addition rule and chain rule

The steps got as

is correct

Next step is to group all dy/dx terms together

is also right step

Now rewrite left side as

Thus both the steps and answer are correct.

the quotient is 4x^2 - 27x + 167 with a remainder of -996

step-by-step explanation:

in synthetic division, use -6 as the divisor and use the coefficients {4, -3, 5, 6}:

-6     )   4     -3     5     6

                -24   162 -1002

          4   -27   167   -996

thus, the quotient is 4x^2 - 27x + 167 with a remainder of -996.

note:   your "4x°-3x+5x+6" is ambiguous / incomplete.   i had to assume that you meant '4x^3 - 3x^2 + 5x + 6."