What is the derivative of arctan(e^{3x}) ?

The derivative of arctan(e^{3x}) is (3e^{3x})/(e^{6x)+1}

What is the first derivative of arctan(e^{3x}) ?

The first derivative of arctan(e^{3x}) is (3e^{3x})/(e^{6x)+1}

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\frac{d}{d x} (arctan (e^{3 x}))

\frac{3 e ^{3 x}}{e ^{6 x} + 1}

Solution steps

\frac{d}{d x} (arctan (e^{3 x}))

Apply the chain rule:

\frac{1}{( e ^{3 x} ) ^{2} + 1} \frac{d}{d x} (e^{3 x})

= \frac{1}{( e ^{3 x} ) ^{2} + 1} \frac{d}{d x} (e^{3 x})

\frac{d}{d x} (e^{3 x}) = e^{3 x} \cdot 3

= \frac{1}{( e ^{3 x} ) ^{2} + 1} e^{3 x} \cdot 3

Simplify

\frac{1}{( e ^{3 x} ) ^{2} + 1} e^{3 x} \cdot 3 : \frac{3 e ^{3 x}}{e ^{6 x} + 1}

= \frac{3 e ^{3 x}}{e ^{6 x} + 1}

What is the derivative of arctan(e^{3x}) ?
The derivative of arctan(e^{3x}) is (3e^{3x})/(e^{6x)+1}
What is the first derivative of arctan(e^{3x}) ?
The first derivative of arctan(e^{3x}) is (3e^{3x})/(e^{6x)+1}