What is the derivative of arccsc(e^x) ?

The derivative of arccsc(e^x) is -1/(sqrt(e^{2x)-1)}

What is the first derivative of arccsc(e^x) ?

The first derivative of arccsc(e^x) is -1/(sqrt(e^{2x)-1)}

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\frac{d}{d x} (arccsc (e^{x}))

- \frac{1}{e ^{2 x} - 1}

Solution steps

\frac{d}{d x} (arccsc (e^{x}))

Apply the chain rule:

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

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

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

= - \frac{1}{( e ^{x} ) ^{2} ( e ^{x} ) ^{2} - 1} e^{x}

Simplify

- \frac{1}{( e ^{x} ) ^{2} ( e ^{x} ) ^{2} - 1} e^{x} : - \frac{1}{e ^{2 x} - 1}

= - \frac{1}{e ^{2 x} - 1}

What is the derivative of arccsc(e^x) ?
The derivative of arccsc(e^x) is -1/(sqrt(e^{2x)-1)}
What is the first derivative of arccsc(e^x) ?
The first derivative of arccsc(e^x) is -1/(sqrt(e^{2x)-1)}