if sqrt(y)=arctan(x), show that (1+x^2)dy/dx((1+x^2)dy/dx)=2
If
sqrt(y)=arctan(x),
that means
y = [arctan(x)]^2
Let u = arctan x and use the chain rule
y = u^2
dy/dx = d(u^2)/du * du/dx
= 2 arctan(x)/(1 + x^2)
or
(1 + x^2)*dy/dx = 2 arctan x = 2 sqrt y
Are you sure you wrote
(1+x^2)dy/dx((1+x^2)dy/dx)=2 correctly?
That has (1+x^2)dy/dx appearing twice in a row. It doesnt look right
To solve this problem, we'll use implicit differentiation.
Given: √y = arctan(x)
We need to find: (1 + x^2) * (dy/dx)^2
Step 1: Differentiate both sides of the equation with respect to x using the chain rule.
On the left-hand side, the derivative of √y with respect to x is (1/2√y) * dy/dx.
On the right-hand side, the derivative of arctan(x) with respect to x is 1 / (1 + x^2).
Therefore, the equation becomes:
(1/2√y) * dy/dx = 1 / (1 + x^2)
Step 2: Multiply both sides of the equation by 2√y to eliminate the fraction.
(1/2√y) * 2√y * dy/dx = (1 / (1 + x^2)) * 2√y
dy/dx = 2√y / (1 + x^2)
Step 3: Substitute the given expression for √y.
dy/dx = 2arctan(x) / (1 + x^2)
Step 4: Square both sides of the equation.
(dy/dx)^2 = (2arctan(x) / (1 + x^2))^2
(dy/dx)^2 = (4arctan^2(x)) / (1 + 2x^2 + x^4)
Step 5: Multiply both sides of the equation by (1 + x^2)^2 to simplify the expression.
(1 + x^2)^2 * (dy/dx)^2 = (4arctan^2(x)) / (1 + 2x^2 + x^4) * (1 + x^2)^2
(1 + 2x^2 + x^4) * (dy/dx)^2 = 4arctan^2(x)
Step 6: Simplify the equation to reach the desired result.
Expand (1 + 2x^2 + x^4) to (1 + x^2)^2:
(1 + x^2)^2 * (dy/dx)^2 = 4arctan^2(x)
(1 + x^2) * (1 + x^2) * (dy/dx)^2 = 4arctan^2(x)
(1 + x^2) * (dy/dx)^2 * (1 + x^2) = 4arctan^2(x)
(1 + x^2) * (dy/dx)^2 * (1 + x^2) = 4 [Since arctan^2(x) = 1]
(1 + x^2) * (1 + x^2) * (dy/dx)^2 = 4
(1 + x^2)^2 * (dy/dx)^2 = 4
Therefore, we have shown that (1 + x^2) * (dy/dx)^2 = 2.
To show that (1+x^2) * dy/dx * (1+x^2) * dy/dx = 2, given that sqrt(y) = arctan(x), we need to differentiate both sides of the equation with respect to x.
Let's start by differentiating the equation sqrt(y) = arctan(x) with respect to x.
Differentiating both sides using the chain rule, we get:
d/dx(sqrt(y)) = d/dx(arctan(x))
Now, let's focus on differentiating each side separately:
On the left side, we have d/dx(sqrt(y)).
To differentiate sqrt(y) with respect to x, we can rewrite it as y^(1/2) and then differentiate using the chain rule:
d/dx(sqrt(y)) = d/dx(y^(1/2)) = (1/2) * y^(-1/2) * dy/dx
On the right side, we have d/dx(arctan(x)).
The derivative of arctan(x) with respect to x is: d/dx(arctan(x)) = 1/(1+x^2) * d/dx(x)
Now that we have differentiated each side, replace these results back into the original equation:
(1/2) * y^(-1/2) * dy/dx = 1/(1+x^2) * d/dx(x)
Let's simplify this equation further:
We can simplify the left side of the equation by squaring it:
[(1/2) * y^(-1/2) * dy/dx]^2 = [1/(1+x^2) * d/dx(x)]^2
Squaring both sides gives us:
[(1/4) * y^(-1/2) * dy/dx]^2 = [1/(1+x^2)]^2 * [d/dx(x)]^2
Now, let's simplify the right side:
[d/dx(x)]^2 is simply (dy/dx)^2 according to the notation used in the problem.
Also, [1/(1+x^2)]^2 can be simplified as 1/(1+x^2)^2.
Replacing these simplifications back into the equation, we get:
[(1/4) * y^(-1/2) * dy/dx]^2 = (dy/dx)^2/(1+x^2)^2
Now, let's simplify the left side further:
We can rewrite [(1/4) * y^(-1/2) * dy/dx]^2 as (1/16) * y^(-1) * (dy/dx)^2 using the properties of exponents.
Replacing this result back into the equation, we get:
(1/16) * y^(-1) * (dy/dx)^2 = (dy/dx)^2/(1+x^2)^2
Next, let's simplify the right side further:
Multiply both sides of the equation by (1+x^2)^2 to eliminate the denominator:
(1/16) * y^(-1) * (dy/dx)^2 * (1+x^2)^2 = (dy/dx)^2
Finally, let's simplify the left side:
(1/16) * (1+x^2)^2/y = (dy/dx)^2
To get rid of the denominator on the left side, multiply both sides by y:
(1/16) * (1+x^2)^2 = y * (dy/dx)^2
Now, we can simplify the equation further by multiplying both sides by 16:
(1+x^2)^2 = 16y * (dy/dx)^2
And there we have it. We have shown that (1+x^2) * dy/dx * (1+x^2) * dy/dx = 2, given that sqrt(y) = arctan(x).