What is the derivative of arccos(e^{5x}) ?

The derivative of arccos(e^{5x}) is -(5e^{5x})/(sqrt(1-e^{10x))}

What is the first derivative of arccos(e^{5x}) ?

The first derivative of arccos(e^{5x}) is -(5e^{5x})/(sqrt(1-e^{10x))}

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\frac{d}{d x} (arccos (e^{5 x}))

- \frac{5 e ^{5 x}}{1 - e ^{10 x}}

Solution steps

\frac{d}{d x} (arccos (e^{5 x}))

Apply the chain rule:

- \frac{1}{1 - ( e ^{5 x} ) ^{2}} \frac{d}{d x} (e^{5 x})

= - \frac{1}{1 - ( e ^{5 x} ) ^{2}} \frac{d}{d x} (e^{5 x})

\frac{d}{d x} (e^{5 x}) = e^{5 x} \cdot 5

= - \frac{1}{1 - ( e ^{5 x} ) ^{2}} e^{5 x} \cdot 5

Simplify

- \frac{1}{1 - ( e ^{5 x} ) ^{2}} e^{5 x} \cdot 5 : - \frac{5 e ^{5 x}}{1 - e ^{10 x}}

= - \frac{5 e ^{5 x}}{1 - e ^{10 x}}

What is the derivative of arccos(e^{5x}) ?
The derivative of arccos(e^{5x}) is -(5e^{5x})/(sqrt(1-e^{10x))}
What is the first derivative of arccos(e^{5x}) ?
The first derivative of arccos(e^{5x}) is -(5e^{5x})/(sqrt(1-e^{10x))}