How do you differentiate #f(x) = sqrt(arctan(3x) # using the chain rule?

Question

How do you differentiate #f(x) = sqrt(arctan(3x)     # using the chain rule?

Ezekiel Alexander · Accepted Answer

Chain rule says that #d/dxf[g(x)]=f'[g(x)]g'(x)#.
Here we have three functions that are #3x#, #arctan# and #sqrt#.
We start from the most "external" that is the square root
We know that #d/dxsqrt(x)=1/(2sqrt(x))# then, applying the chain rule we have #d/dxsqrt(arctan(3x))=1/(2sqrt(arctan(3x)))d/dxarctan(3x)#. We have now to calculate the derivative of #arctan(3x)# We know that #d/dxarctan(x)=1/(1+x^2)# then we reapply the chain rule #d/dxarctan(3x)=1/(1+(3x)^2)d/dx3x# and finally the easy one #d/dx3x=3#. We substitute everything back and write #d/dxsqrt(arctan(3x))=3/(2sqrt(arctan(3x))(1+9x^2)#

Elijah Abram · Answer

undefined To differentiate $ f(x) = \sqrt{\arctan(3x)} $ using the chain rule, follow these steps:

1. Identify the outer function and the inner function.
2. Differentiate the outer function with respect to the inner function.
3. Differentiate the inner function with respect to $ x $.
4. Multiply the results of steps 2 and 3 together to find the derivative.

Let's differentiate step by step:

Given function: $ f(x) = \sqrt{\arctan(3x)} $

1. Outer function: $ \sqrt{x} $
   Inner function: $ \arctan(3x) $

2. Differentiate the outer function with respect to the inner function:
   $ \frac{d}{du} \sqrt{u} = \frac{1}{2\sqrt{u}} $

3. Differentiate the inner function with respect to $ x $:
   $ \frac{d}{dx} \arctan(3x) = \frac{3}{1 + (3x)^2} $

4. Multiply the results of steps 2 and 3 together:
   $ \frac{1}{2\sqrt{\arctan(3x)}} \times \frac{3}{1 + (3x)^2} $

Thus, the derivative of $ f(x) = \sqrt{\arctan(3x)} $ using the chain rule is:
   $ f'(x) = \frac{3}{2\sqrt{(1 + (3x)^2) \cdot \arctan(3x)}} $