If 4x2+3x+xy=2 and y(2)=–10 , find y'(2) by implicit differentiation.
i keep getting the wrong answer for this problem evn when plugging 2 in for y.
you can try this....
so f'(x)= -8x-3/(1+ xy)
Plug in 10 for x and 2 for y
(-8(-10)-3)/(1+(-10)(2)= -3.66667
how do u get the - 8x over one?
To find the derivative of y with respect to x, denoted as y', using implicit differentiation, we need to differentiate both sides of the equation with respect to x. Let's go step by step.
Step 1: Differentiate both sides of the equation with respect to x.
Take the derivative of each term on both sides. The left-hand side of the equation will involve the product rule, as it contains the xy term.
On the left-hand side:
d/dx (4x²) = 8x
d/dx (3x) = 3
d/dx (xy) = x(dy/dx) + y
On the right-hand side, the constant term 2 differentiates to zero.
So we have: 8x + 3 + x(dy/dx) + y = 0
Step 2: Simplify the equation.
Rearrange the equation to isolate the term dy/dx, which represents y'.
x(dy/dx) + y = -8x - 3
x(dy/dx) = -y - 8x - 3
Step 3: Substitute the given value.
Plug in the particular value y(2) = -10 when x = 2.
2(dy/dx) = -(-10) - 8(2) - 3
2(dy/dx) = 10 - 16 - 3
2(dy/dx) = -9
(dy/dx) = -9/2
Therefore, y'(2) (the derivative of y with respect to x when x = 2) is -9/2.
Make sure to double-check your calculations to avoid any arithmetic errors.