If x2+xy=10, then when x=2, dy/dx=?
To find dy/dx when x = 2, we need to differentiate both sides of the equation x^2 + xy = 10 with respect to x.
Taking the derivative of both sides, we use the product rule for differentiating xy:
d/dx (x^2 + xy) = d/dx (10)
2x + x(dy/dx) + y = 0
Now, we substitute x = 2 into the equation:
2(2) + 2(dy/dx) + y = 0
4 + 2(dy/dx) + y = 0
Rearranging the equation to isolate dy/dx:
2(dy/dx) = -4 - y
dy/dx = (-4 - y) / 2
So, when x = 2, dy/dx = (-4 - y) / 2.
To find dy/dx when x=2, we need to differentiate the given equation with respect to x using the rules of differentiation.
Start by differentiating both sides of the equation:
d/dx (x^2 + xy) = d/dx (10)
To differentiate the left side, we apply the power rule and the product rule.
Differentiating x^2 using the power rule:
d/dx (x^2) = 2x
Now, differentiating the product xy using the product rule:
d/dx (xy) = x(dy/dx) + y
We can rewrite the equation as:
2x + x(dy/dx) + y = 0
Next, substitute the given value x = 2 into the equation:
2(2) + 2(dy/dx) + y = 0
Simplifying the equation:
4 + 2(dy/dx) + y = 0
To find dy/dx when x = 2, we need to find the value of y. To do that, we can substitute x = 2 into the original equation:
(2^2) + (2y) = 10
4 + 2y = 10
2y = 10 - 4
2y = 6
y = 6/2
y = 3
Now substitute the values x = 2 and y = 3 back into the equation:
4 + 2(dy/dx) + 3 = 0
Simplifying the equation further:
7 + 2(dy/dx) = 0
Isolating dy/dx, we subtract 7 from both sides:
2(dy/dx) = -7
Finally, divide both sides by 2 to solve for dy/dx:
dy/dx = -7/2
Therefore, when x = 2, dy/dx is equal to -7/2.
2x dx + x dy + y dx = 0
x dy = -(2x+y) dy
dy/dx = - x/(2x+y)
when x = 2, y = (10-x^2)/x = 3
so
-2/(4+3) = -2/7