What is the derivative of \sqrt[3]{sec(5x}) ?

The derivative of \sqrt[3]{sec(5x}) is (5tan(5x)sec^{1/3}(5x))/3

What is the first derivative of \sqrt[3]{sec(5x}) ?

The first derivative of \sqrt[3]{sec(5x}) is (5tan(5x)sec^{1/3}(5x))/3

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\frac{5 tan ( 5 x ) sec ^{\frac{1}{3}} ( 5 x )}{3}

Solution steps

Apply the chain rule:

\frac{1}{3 ( sec ( 5 x ) ) ^{\frac{2}{3}}} \frac{d}{d x} (sec (5 x))

= \frac{1}{3 ( sec ( 5 x ) ) ^{\frac{2}{3}}} \frac{d}{d x} (sec (5 x))

\frac{d}{d x} (sec (5 x)) = sec (5 x) tan (5 x) \cdot 5

= \frac{1}{3 ( sec ( 5 x ) ) ^{\frac{2}{3}}} sec (5 x) tan (5 x) \cdot 5

Simplify

\frac{1}{3 sec ^{\frac{2}{3}} ( 5 x )} sec (5 x) tan (5 x) \cdot 5 : \frac{5 tan ( 5 x ) sec ^{\frac{1}{3}} ( 5 x )}{3}

= \frac{5 tan ( 5 x ) sec ^{\frac{1}{3}} ( 5 x )}{3}

What is the derivative of \sqrt[3]{sec(5x}) ?
The derivative of \sqrt[3]{sec(5x}) is (5tan(5x)sec^{1/3}(5x))/3
What is the first derivative of \sqrt[3]{sec(5x}) ?
The first derivative of \sqrt[3]{sec(5x}) is (5tan(5x)sec^{1/3}(5x))/3