What is the derivative of cos^3(sqrt(x^2-1)) ?

The derivative of cos^3(sqrt(x^2-1)) is -(3xcos^2(sqrt(x^2-1))sin(sqrt(x^2-1)))/(sqrt(x^2-1))

What is the first derivative of cos^3(sqrt(x^2-1)) ?

The first derivative of cos^3(sqrt(x^2-1)) is -(3xcos^2(sqrt(x^2-1))sin(sqrt(x^2-1)))/(sqrt(x^2-1))

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\frac{d}{d x} (cos^{3} (x^{2} - 1))

- \frac{3 x cos ^{2} ( x ^{2} - 1 ) sin ( x ^{2} - 1 )}{x ^{2} - 1}

Solution steps

\frac{d}{d x} (cos^{3} (x^{2} - 1))

Apply the chain rule:

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

= 3 (cos (x^{2} - 1))^{2} \frac{d}{d x} (cos (x^{2} - 1))

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

= 3 (cos (x^{2} - 1))^{2} (- \frac{x sin ( x ^{2} - 1 )}{x ^{2} - 1})

Simplify

3 cos^{2} (x^{2} - 1) (- \frac{x sin ( x ^{2} - 1 )}{x ^{2} - 1}) : - \frac{3 x cos ^{2} ( x ^{2} - 1 ) sin ( x ^{2} - 1 )}{x ^{2} - 1}

= - \frac{3 x cos ^{2} ( x ^{2} - 1 ) sin ( x ^{2} - 1 )}{x ^{2} - 1}

What is the derivative of cos^3(sqrt(x^2-1)) ?
The derivative of cos^3(sqrt(x^2-1)) is -(3xcos^2(sqrt(x^2-1))sin(sqrt(x^2-1)))/(sqrt(x^2-1))
What is the first derivative of cos^3(sqrt(x^2-1)) ?
The first derivative of cos^3(sqrt(x^2-1)) is -(3xcos^2(sqrt(x^2-1))sin(sqrt(x^2-1)))/(sqrt(x^2-1))