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How do you find the derivative #h(x)=x^2arctanx#?

Answer 1

Christopher Allers

#h'(x) = x^2/(x^2+1)+2xarctan(x)#

#h(x) = x^2arctan(x)#

Apply the product rule.

#h'(x) = x^2 * d/dx arctan(x) + d/dx x^2 * arctan(x)#

Apply standard differential and power rule.

#h'(x) = x^2 * 1/(x^2+1) + 2x * arctan(x)#

#= x^2/(x^2+1)+2xarctan(x)#

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Answer 2

Luke Alvarez

To find the derivative of ( h(x) = x^2 \arctan(x) ), you would use the product rule.

[ h'(x) = \frac{d}{dx}(x^2) \arctan(x) + x^2 \frac{d}{dx}(\arctan(x)) ]

Using the power rule, ( \frac{d}{dx}(x^2) = 2x ), and using the derivative of ( \arctan(x) ), which is ( \frac{1}{1+x^2} ), we get:

[ h'(x) = 2x \arctan(x) + \frac{x^2}{1+x^2} ]

Therefore, the derivative of ( h(x) = x^2 \arctan(x) ) is ( 2x \arctan(x) + \frac{x^2}{1+x^2} ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.