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How do you find the derivative of #arctan(x^2y)#?

Answer 1

#d/dx(arctan(x^2y)) = (2xy)/(1 + (x^2y)^2)#

So, basically, you want to find #d/dx(arctan(x^2y))#.

We need to first observe that #y# and #x# have no relation to each other in the expression. This observation is very important, since now #y# can be treated as a constant with respect to #x#.

We first apply chain rule: #d/dx(arctan(x^2y)) = d/(d(x^2y))(arctan(x^2y)) xx d/dx(x^2y) = 1/(1 + (x^2y)^2) xx d/dx(x^2y)#.

Here, as we mentioned earlier, #y# is a constant with respect to #x#. So,

#d/dx(x^2 color(red)(y)) = color(red)(y) xx d/dx(x^2) = 2xy#

So, #d/dx(arctan(x^2y)) = 1/(1 + (x^2y)^2) xx 2xy = (2xy)/(1 + (x^2y)^2)#

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Answer 2

Charles Anctil

To find the derivative of arctan(x^2y) with respect to x, you would use the chain rule and implicit differentiation. The derivative would be (1/(1 + (x^2y)^2)) * (2xy + 2x^3y^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.