find the derivative of y=sqrt of arctan x
y'=[(1/2)*(arctan(x))^(-1/2)]*(1/(1+x^(2)))
Use the chain rule d/dx[f(x)^(n)]=
(n(f(x))^(n-1))*(f'(x)).
To find the derivative of \(y = \sqrt{\arctan x}\), we can use the chain rule. The chain rule states that if we have a composite function, such as \(y = f(g(x))\), then the derivative of \(y\) with respect to \(x\) can be found by multiplying the derivative of \(f\) with respect to \(g\) by the derivative of \(g\) with respect to \(x\).
Let's break down the steps to find the derivative of \(y = \sqrt{\arctan x}\):
Step 1: Define the functions:
Let \(f(u) = \sqrt{u}\) and \(g(x) = \arctan x\).
Step 2: Find the derivatives:
To find the derivative of \(f(u) = \sqrt{u}\), we can use the power rule: \(\frac{d}{du} \sqrt{u} = \frac{1}{2\sqrt{u}}\).
To find the derivative of \(g(x) = \arctan x\), we can use the chain rule: \(\frac{d}{dx} \arctan x = \frac{1}{1+x^2}\) (the derivative of \(\arctan x\) is \(\frac{1}{1+x^2}\)).
Step 3: Apply the chain rule:
Using the chain rule, the derivative of \(y = f(g(x))\) is given by \(\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\).
Applying this to our functions, we have:
\(\frac{dy}{dx} = \frac{1}{2\sqrt{g(x)}} \cdot \frac{1}{1+x^2}\).
Step 4: Simplify the expression:
Since \(g(x) = \arctan x\), we can substitute into the expression:
\(\frac{dy}{dx} = \frac{1}{2\sqrt{\arctan x}} \cdot \frac{1}{1+x^2}\).
This is the derivative of \(y = \sqrt{\arctan x}\).