What is the derivative of x*arcsin(sqrt(x)) ?

The derivative of x*arcsin(sqrt(x)) is arcsin(sqrt(x))+(sqrt(x))/(2sqrt(1-x))

What is the first derivative of x*arcsin(sqrt(x)) ?

The first derivative of x*arcsin(sqrt(x)) is arcsin(sqrt(x))+(sqrt(x))/(2sqrt(1-x))

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derivative

arcsin (x) + \frac{x}{2 1 - x}

Solution steps

\frac{d}{d x} (x arcsin (x))

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = arcsin (x)

= \frac{d x}{d x} arcsin (x) + \frac{d}{d x} (arcsin (x)) x

\frac{d x}{d x} = 1

\frac{d}{d x} (arcsin (x)) = \frac{1}{2 x 1 - x}

= 1 \cdot arcsin (x) + \frac{1}{2 x 1 - x} x

Simplify

1 \cdot arcsin (x) + \frac{1}{2 x 1 - x} x : arcsin (x) + \frac{x}{2 1 - x}

= arcsin (x) + \frac{x}{2 1 - x}

What is the derivative of x*arcsin(sqrt(x)) ?
The derivative of x*arcsin(sqrt(x)) is arcsin(sqrt(x))+(sqrt(x))/(2sqrt(1-x))
What is the first derivative of x*arcsin(sqrt(x)) ?
The first derivative of x*arcsin(sqrt(x)) is arcsin(sqrt(x))+(sqrt(x))/(2sqrt(1-x))