How do you find the derivative of #(arctan x)^3#?

Question

Evelyn Agard · Accepted Answer

You can find it like this:

#f(x)=arctan^3(x)# #(d[arctan(x)])/(dx)=1/((x^2+1))# So you apply the chain rule: #f'(x)=3arctan^2(x)xx1/((x^2+1))# #f'(x)=(3arctan^2(x))/((x^2+1))#

Evelyn Agard · Answer

undefined The derivative of (arctan x)^3 with respect to x is 3*(arctan x)^2 / (1 + x^2).

Aurora Alvarado · Answer

undefined To find the derivative of $ (\arctan x)^3 $, we can use the chain rule of differentiation. Let's denote $ u = \arctan x $. Then, we have $ y = u^3 $. Using the chain rule, the derivative of $ y $ with respect to $ x $ is:

$$ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} $$

First, we find $ \frac{dy}{du} $ by applying the power rule to $ u^3 $, which gives us:

$$ \frac{dy}{du} = 3u^2 $$

Next, we find $ \frac{du}{dx} $, the derivative of $ \arctan x $ with respect to $ x $. This derivative is $ \frac{1}{1+x^2} $ by the derivative of the arctan function.

Now, we multiply these two derivatives together:

$$ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 3u^2 \cdot \frac{1}{1+x^2} $$

Finally, substituting $ u = \arctan x $ back into the expression, we get:

$$ \frac{dy}{dx} = 3(\arctan x)^2 \cdot \frac{1}{1+x^2} $$

So, the derivative of $ (\arctan x)^3 $ with respect to $ x $ is $ 3(\arctan x)^2 \cdot \frac{1}{1+x^2} $.