What is (\partial)/(\partial x)(xsqrt(1-x^2)) ?

The answer to (\partial)/(\partial x)(xsqrt(1-x^2)) is (-2x^2+1)/(sqrt(1-x^2))

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(\partial)/(\partial x)(xsqrt(1-x^2))

\frac{\partial}{\partial x} (x 1 - x^{2})

\frac{- 2 x ^{2} + 1}{1 - x ^{2}}

Solution steps

\frac{\partial}{\partial x} (x 1 - x^{2})

Apply the Product Rule:

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

f = x, g = 1 - x^{2}

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

\frac{\partial}{\partial x} (x) = 1

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

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

Simplify

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

= \frac{- 2 x ^{2} + 1}{1 - x ^{2}}

y = 8 x, y = 0, y = x^{2} - 9

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\frac{\partial}{\partial y} (2 x + y - 8)

What is (\partial)/(\partial x)(xsqrt(1-x^2)) ?
The answer to (\partial)/(\partial x)(xsqrt(1-x^2)) is (-2x^2+1)/(sqrt(1-x^2))