What is the derivative of y=(x^3+x)arcsin(x) ?

The derivative of y=(x^3+x)arcsin(x) is arcsin(x)(3x^2+1)+(x^3+x)/(sqrt(1-x^2))

What is the first derivative of y=(x^3+x)arcsin(x) ?

The first derivative of y=(x^3+x)arcsin(x) is arcsin(x)(3x^2+1)+(x^3+x)/(sqrt(1-x^2))

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derivative

arcsin (x) (3 x^{2} + 1) + \frac{x ^{3} + x}{1 - x ^{2}}

Solution steps

\frac{d}{d x} ((x^{3} + x) arcsin (x))

Apply the Product Rule:

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

f = x^{3} + x, g = arcsin (x)

= \frac{d}{d x} (x^{3} + x) arcsin (x) + \frac{d}{d x} (arcsin (x)) (x^{3} + x)

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

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

= (3 x^{2} + 1) arcsin (x) + \frac{1}{1 - x ^{2}} (x^{3} + x)

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

= arcsin (x) (3 x^{2} + 1) + \frac{x ^{3} + x}{1 - x ^{2}}

What is the derivative of y=(x^3+x)arcsin(x) ?
The derivative of y=(x^3+x)arcsin(x) is arcsin(x)(3x^2+1)+(x^3+x)/(sqrt(1-x^2))
What is the first derivative of y=(x^3+x)arcsin(x) ?
The first derivative of y=(x^3+x)arcsin(x) is arcsin(x)(3x^2+1)+(x^3+x)/(sqrt(1-x^2))