What is the derivative of asin^3(x/3) ?

The derivative of asin^3(x/3) is asin^2(x/3)cos(x/3)

What is the first derivative of asin^3(x/3) ?

The first derivative of asin^3(x/3) is asin^2(x/3)cos(x/3)

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\frac{d}{d x} (a sin^{3} (\frac{x}{3}))

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

Solution steps

\frac{d}{d x} (a sin^{3} (\frac{x}{3}))

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= a \frac{d}{d x} (sin^{3} (\frac{x}{3}))

Apply the chain rule:

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

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

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

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

Simplify

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

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

What is the derivative of asin^3(x/3) ?
The derivative of asin^3(x/3) is asin^2(x/3)cos(x/3)
What is the first derivative of asin^3(x/3) ?
The first derivative of asin^3(x/3) is asin^2(x/3)cos(x/3)