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

The derivative of sqrt(cos^3(x^2)) is -3xsin(x^2)sqrt(cos(x^2))

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

The first derivative of sqrt(cos^3(x^2)) is -3xsin(x^2)sqrt(cos(x^2))

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

- 3 x sin (x^{2}) cos (x^{2})

Solution steps

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

Apply the chain rule:

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

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

\frac{d}{d x} (cos^{3} (x^{2})) = - 6 x cos^{2} (x^{2}) sin (x^{2})

= \frac{1}{2 cos ^{3} ( x ^{2} )} (- 6 x cos^{2} (x^{2}) sin (x^{2}))

Simplify

\frac{1}{2 cos ^{3} ( x ^{2} )} (- 6 x cos^{2} (x^{2}) sin (x^{2})) : - 3 x sin (x^{2}) cos (x^{2})

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

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