What is the derivative of x(1-x^2^{1/2}) ?

The derivative of x(1-x^2^{1/2}) is (-2x^2+1)/((1-x^2)^{1/2)}

What is the first derivative of x(1-x^2^{1/2}) ?

The first derivative of x(1-x^2^{1/2}) is (-2x^2+1)/((1-x^2)^{1/2)}

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

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

Solution steps

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

Apply the Product Rule:

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

f = x, g = (1 - x^{2})^{\frac{1}{2}}

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

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

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

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

Simplify

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

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

n = 0 \sum \infty i (- 1)^{n}

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

\frac{d}{d x} (4 arccos (1 - x^{2}))

\frac{\partial}{\partial y} (- 2 y^{2})

What is the derivative of x(1-x^2^{1/2}) ?
The derivative of x(1-x^2^{1/2}) is (-2x^2+1)/((1-x^2)^{1/2)}
What is the first derivative of x(1-x^2^{1/2}) ?
The first derivative of x(1-x^2^{1/2}) is (-2x^2+1)/((1-x^2)^{1/2)}