if x ^ 2 + xy = 10 ,thn when x = 2,d/dx (y) is ?
of course, since y=3 when x=2,
dy/dx = -2 - 3/2 = -7/2
Apologies for the mistake. Thank you for catching it.
Since y = 3 when x = 2, we can substitute these values into the equation for dy/dx:
(dy/dx) = -2 - (y/2)
(dy/dx) = -2 - (3/2)
(dy/dx) = -2 - 1.5
(dy/dx) = -3.5
Therefore, when x = 2 and y = 3, the value of dy/dx is -3.5.
To find d/dx (y) when x = 2, we need to differentiate both sides of the equation x^2 + xy = 10 with respect to x.
Differentiating x^2 with respect to x gives 2x.
To differentiate xy, we need to use the product rule. The product rule states that d(uv)/dx = u * dv/dx + v * du/dx.
In this case, u = x and v = y.
So, using the product rule, we get d(xy)/dx = x * d(y)/dx + y * d(x)/dx.
Since the equation is x^2 + xy = 10, we can substitute the expression for d(xy)/dx back in:
2x + x * d(y)/dx + y * d(x)/dx = 0.
Now, let's substitute x = 2 into this equation:
2(2) + 2 * d(y)/dx + y * d(2)/dx = 0.
Simplifying further:
4 + 2 * d(y)/dx + 2y = 0.
Rearranging the equation to isolate d(y)/dx:
2 * d(y)/dx = -4 - 2y.
Finally, dividing both sides of the equation by 2:
d(y)/dx = (-4 - 2y) / 2.
So, when x = 2, d/dx (y) is (-4 - 2y) / 2.
To find the derivative of y with respect to x, we need to differentiate the given equation with respect to x using implicit differentiation. Let's go through the steps:
1. Start with the given equation: x^2 + xy = 10.
2. Take the derivative of both sides of the equation with respect to x. Treat y as a function of x and apply the chain rule. Let's differentiate each term separately:
- For the term x^2, the derivative is 2x.
- For the term xy, we need to apply the product rule. The derivative of xy with respect to x is y + x * dy/dx.
Therefore, the differentiated equation becomes: 2x + y + x * dy/dx = 0.
3. Solve the differentiated equation for dy/dx. Rearrange the equation to isolate dy/dx:
dy/dx = -2x - y / x.
4. Substitute the given value of x = 2 into the equation above to find the required derivative:
dy/dx = -2(2) - y / 2.
5. Simplify the expression:
dy/dx = -4 - y / 2.
Thus, when x = 2, the derivative of y with respect to x is -4 - y / 2.
To find the derivative of y with respect to x, we can differentiate the equation x^2 + xy = 10 with respect to x using the product rule and chain rule.
Differentiating both sides with respect to x:
2x + x(dy/dx) + y = 0
Rearranging the equation and isolating dy/dx:
x(dy/dx) = -2x - y
(dy/dx) = (-2x - y) / x
Substituting x = 2 into the equation for dy/dx:
(dy/dx) = (-2(2) - y) / 2
(dy/dx) = (-4 - y) / 2
(dy/dx) = -2 - (y/2)
Therefore, when x = 2, d/dx (y) is -2 - (y/2).