what is the 2nd derivative of:
1-xy = x-y
i get (-y-1)/(x-1) for the first
the second i know is (-y-1)/(x-1)^2 but i keep getting (0)/(x-1)^2
please show me the steps i should be taking and if possible what i did wrong
I would solve it for y first
1-xy = x-y
y - xy = x-1
y(1 - x) = x-1
y = (x-1)/(1-x) = -1 , x ≠1
Well, isn't that special !!!
so y' = 0 and of course
y '' = 0
Wolfram took it as a pair of intersecting lines.
Notice the horizontal line y = -1
The vertical line x = 1 will give us y = any real number
e.g. (1, 4.56) will work
LS = 1 - 1(4.56)
RS = 1 - 4.56
LS = RS
http://www.wolframalpha.com/input/?i=plot+1-xy+%3D+x-y+
Well, it seems like you're having some trouble finding the correct second derivative. Let's go through the steps together and see where the confusion might be.
Starting with the given equation:
1 - xy = x - y
To find the first derivative, we need to differentiate both sides of the equation with respect to x:
d/dx(1 - xy) = d/dx(x - y)
For the left-hand side, we use the product rule:
d/dx(1 - xy) = d/dx(1) - x(d/dx(y)) - y(d/dx(x))
= 0 - x(dy/dx) - y
For the right-hand side, we have:
d/dx(x - y) = 1 - dy/dx
Setting the two derivatives equal to each other, we get:
-x(dy/dx) - y = 1 - dy/dx
Simplifying the equation, we have:
dy/dx - x(dy/dx) = -y - 1
Factoring out dy/dx on the left-hand side:
(1 - x) dy/dx = -y - 1
Dividing both sides by (1 - x), we find:
dy/dx = (-y - 1)/(1 - x)
So far, so good! Now let's move on to finding the second derivative.
To find the second derivative, we need to differentiate dy/dx with respect to x. Therefore, we differentiate the previously derived equation:
d/dx[(-y - 1)/(1 - x)] = d/dx(dy/dx)
Using the quotient rule, we have:
[(1 - x)(d²y/dx²) - (-y - 1)(d/dx(1 - x))] / (1 - x)² = d²y/dx²
Expanding and simplifying, we get:
[(1 - x)(d²y/dx²) + (y + 1)] / (1 - x)² = d²y/dx²
And there you have it! The second derivative of the given equation is:
d²y/dx² = [(1 - x)(d²y/dx²) + (y + 1)] / (1 - x)²
It appears that you had an error when differentiating the equation. Remember to use the quotient rule and simplify your expression correctly. Keep practicing, and I'm sure you'll master those derivatives in no time.
To find the second derivative of the equation 1 - xy = x - y, we'll need to find the first derivative first. Let's start by finding the first derivative:
1. Differentiate both sides of the equation with respect to x:
d/dx (1 - xy) = d/dx (x - y)
2. Apply the product rule to differentiate the term xy:
(d/dx (1))(1) - (1)(d/dx (xy)) = 1 - dy/dx
3. Simplify the equation:
0 - (x(dy/dx) + y) = 1 - dy/dx
4. Rearrange the equation to solve for dy/dx (the first derivative):
- (x(dy/dx) + y) = 1 - dy/dx
-xy - y = 1 - dy/dx + x(dy/dx)
-xy - y = 1 + (x - 1)(dy/dx)
dy/dx = (1 - xy -y)/(x - 1)
Now we have the first derivative. Let's proceed to find the second derivative:
5. Differentiate the expression dy/dx (which is the first derivative) with respect to x:
d/dx [(1 - xy - y)/(x - 1)]
6. Use the quotient rule to differentiate the expression:
[(x - 1)(d/dx (1 - xy - y)) - (1 - xy - y)(d/dx (x - 1))]/(x - 1)^2
7. Simplify the equation:
[(x - 1)(-y - dy/dx) - (1 - xy - y)(1)]/(x - 1)^2
8. Expand and simplify:
(-xy + y + y + xy + dy/dx - x + 1 -xy -y)/ (x - 1)^2
9. Combine like terms:
(2y - x - 1 + dy/dx)/ (x - 1)^2
Thus, the second derivative of the given equation is:
(2y - x - 1 + dy/dx)/ (x - 1)^2
It seems that you made an error while applying the quotient rule in step 6. Make sure to evaluate each term correctly while differentiating.
To find the second derivative of the equation 1 - xy = x - y, we will follow a step-by-step process.
Step 1: Rewrite the given equation in a more suitable form. We rearrange the terms to get:
xy - y = x - 1
Step 2: Differentiate both sides of the equation with respect to x using the product rule for differentiation. The product rule states that if we have two functions, u(x) and v(x), their product is represented as (u * v). The derivative of the product is given by the formula:
d/dx (u * v) = (d/dx u) * v + u * (d/dx v)
Using the product rule, let's differentiate each term of the equation:
For the left-hand side (LHS):
d/dx (xy - y) = (d/dx xy) - (d/dx y)
Applying the product rule:
= (x * (d/dx y) + y * (d/dx x)) - (d/dx y)
= x * (dy/dx) + y - (dy/dx)
For the right-hand side (RHS):
d/dx (x - 1) = (d/dx x) - (d/dx 1) = 1 - 0 = 1
So, the differentiated equation becomes:
x * (dy/dx) + y - (dy/dx) = 1
Step 3: Simplify the equation by combining like terms:
(x - 1) * (dy/dx) + y = 1
Step 4: Solve the resulting equation for (dy/dx) by isolating the derivative term:
(x - 1) * (dy/dx) = 1 - y
Step 5: Now, differentiate the equation obtained in Step 4 with respect to x again to find the second derivative. We can use the product rule again:
For the left-hand side:
(d/dx) [(x - 1) * (dy/dx)] = (d/dx(x - 1)) * (dy/dx) + (x - 1) * (d^2y/dx^2)
Applying the product rule:
= [(d/dx(x - 1)) * (dy/dx)] + [(x - 1) * (d^2y/dx^2)]
= (1 * (dy/dx)) + (x - 1) * (d^2y/dx^2)
= (dy/dx) + (x - 1) * (d^2y/dx^2)
For the right-hand side:
(d/dx) (1 - y) = d(1)/dx - d(y)/dx
= 0 - (d(y)/dx)
= -(dy/dx)
So, the equation becomes:
(dy/dx) + (x - 1) * (d^2y/dx^2) = -(dy/dx)
Step 6: Simplify the equation by canceling out the common derivative term on both sides:
(x - 1) * (d^2y/dx^2) = -2 * (dy/dx)
Finally, divide both sides by (x - 1) to solve for (d^2y/dx^2):
d^2y/dx^2 = -2 * (dy/dx) / (x - 1)
Therefore, the second derivative of the equation 1 - xy = x - y is:
d^2y/dx^2 = -2 * (dy/dx) / (x - 1)
It seems there was a mistake in your approach. Please review your calculations to identify the error.