What is the derivative of sec^5(sqrt(x^2-1)) ?

The derivative of sec^5(sqrt(x^2-1)) is (5xsec^5(sqrt(x^2-1))tan(sqrt(x^2-1)))/(sqrt(x^2-1))

What is the first derivative of sec^5(sqrt(x^2-1)) ?

The first derivative of sec^5(sqrt(x^2-1)) is (5xsec^5(sqrt(x^2-1))tan(sqrt(x^2-1)))/(sqrt(x^2-1))

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

\frac{5 x sec ^{5} ( x ^{2} - 1 ) tan ( x ^{2} - 1 )}{x ^{2} - 1}

Solution steps

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

Apply the chain rule:

5 (sec (x^{2} - 1))^{4} \frac{d}{d x} (sec (x^{2} - 1))

= 5 (sec (x^{2} - 1))^{4} \frac{d}{d x} (sec (x^{2} - 1))

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

= 5 (sec (x^{2} - 1))^{4} \frac{x sec ( x ^{2} - 1 ) tan ( x ^{2} - 1 )}{x ^{2} - 1}

Simplify

5 sec^{4} (x^{2} - 1) \frac{x sec ( x ^{2} - 1 ) tan ( x ^{2} - 1 )}{x ^{2} - 1} : \frac{5 x sec ^{5} ( x ^{2} - 1 ) tan ( x ^{2} - 1 )}{x ^{2} - 1}

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

3 x^{2} + 1

f (x) = x^{\frac{1}{9}}

f (x) = 2 x (2 - 4 x)^{4}, at x = 1

\int \frac{tan ^{2} ( x )}{sec ^{3} ( x )} d x

What is the derivative of sec^5(sqrt(x^2-1)) ?
The derivative of sec^5(sqrt(x^2-1)) is (5xsec^5(sqrt(x^2-1))tan(sqrt(x^2-1)))/(sqrt(x^2-1))
What is the first derivative of sec^5(sqrt(x^2-1)) ?
The first derivative of sec^5(sqrt(x^2-1)) is (5xsec^5(sqrt(x^2-1))tan(sqrt(x^2-1)))/(sqrt(x^2-1))