What is the derivative of arctan(ax^2) ?

The derivative of arctan(ax^2) is (2ax)/(a^2x^4+1)

What is the first derivative of arctan(ax^2) ?

The first derivative of arctan(ax^2) is (2ax)/(a^2x^4+1)

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\frac{d}{d x} (arctan (a x^{2}))

\frac{2 a x}{a ^{2} x ^{4} + 1}

Solution steps

\frac{d}{d x} (arctan (a x^{2}))

Apply the chain rule:

\frac{1}{( a x ^{2} ) ^{2} + 1} \frac{d}{d x} (a x^{2})

= \frac{1}{( a x ^{2} ) ^{2} + 1} \frac{d}{d x} (a x^{2})

\frac{d}{d x} (a x^{2}) = 2 a x

= \frac{1}{( a x ^{2} ) ^{2} + 1} \cdot 2 a x

Simplify

\frac{1}{( a x ^{2} ) ^{2} + 1} \cdot 2 a x : \frac{2 a x}{a ^{2} x ^{4} + 1}

= \frac{2 a x}{a ^{2} x ^{4} + 1}

What is the derivative of arctan(ax^2) ?
The derivative of arctan(ax^2) is (2ax)/(a^2x^4+1)
What is the first derivative of arctan(ax^2) ?
The first derivative of arctan(ax^2) is (2ax)/(a^2x^4+1)