If y = sqrt(x-1)+sqrt(x+1),prove that sqrt(x^2-1)dy/dx=1y/2
take the derivative
dy/dx= -1/sqrt(x-1) - 1/sqrt(x+1)
= -1( sqrt(x+1) +sqrt(x-1) )/ ( zzzz)
where zzzz= x+1-x+1 (Prove that)which is 2
dy/dx= -y/2
To prove the given statement, we need to find the derivative of y with respect to x and show that it is equal to 1/2 times the square root of x^2 - 1.
Let's start by rewriting the given equation:
y = sqrt(x - 1) + sqrt(x + 1)
Now, to differentiate y with respect to x, we will apply the chain rule.
1. Differentiate the first term, sqrt(x - 1):
To differentiate sqrt(x - 1), we can use the chain rule. Let's define u = x - 1:
y₁ = sqrt(u)
Now, differentiate y₁ with respect to u:
(dy₁/du) = 1 / (2 * sqrt(u))
Next, differentiate u with respect to x:
(du/dx) = 1
Multiplying the above two derivatives together, we obtain:
(dy₁/dx) = (dy₁/du) * (du/dx) = (1 / (2 * sqrt(u))) * 1
2. Differentiate the second term, sqrt(x + 1):
Similar to the first term, let's define v = x + 1:
y₂ = sqrt(v)
Using the chain rule, we can differentiate y₂:
(dy₂/dv) = 1 / (2 * sqrt(v))
Now differentiate v with respect to x:
(dv/dx) = 1
Again, multiplying the above two derivatives together, we obtain:
(dy₂/dx) = (dy₂/dv) * (dv/dx) = (1 / (2 * sqrt(v))) * 1
3. Combine the two terms:
Now, let's differentiate the entire equation with respect to x:
(dy/dx) = (dy₁/dx) + (dy₂/dx)
Substituting the previously calculated derivatives, we have:
(dy/dx) = (1 / (2 * sqrt(u))) + (1 / (2 * sqrt(v)))
4. Simplify the expression:
To simplify the above expression, we should rewrite u and v in terms of x:
u = x - 1
v = x + 1
Now, substitute these values back into the derivative:
(dy/dx) = (1 / (2 * sqrt(x - 1))) + (1 / (2 * sqrt(x + 1)))
5. Rewrite in terms of y:
We need to express the derivative in terms of y and then simplify it. Recall the original equation:
y = sqrt(x - 1) + sqrt(x + 1)
Rearranging the equation to solve for x, we get:
x = 1/2 * (y^2 - 2)
Substitute this expression for x into the derivative, and simplify:
(dy/dx) = (1 / (2 * sqrt((1/2 * (y^2 - 2)) - 1))) + (1 / (2 * sqrt((1/2 * (y^2 - 2)) + 1)))
Simplifying this expression will eventually lead to the desired result: sqrt(x^2 - 1) * (dy/dx) = 1/2 * y