What is the derivative of-x(1+x^2^{-3/2}) ?

The derivative of-x(1+x^2^{-3/2}) is -(-2x^2+1)/((x^2+1)^{5/2)}

What is the first derivative of-x(1+x^2^{-3/2}) ?

The first derivative of-x(1+x^2^{-3/2}) is -(-2x^2+1)/((x^2+1)^{5/2)}

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

- \frac{- 2 x ^{2} + 1}{( x ^{2} + 1 ) ^{\frac{5}{2}}}

Solution steps

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

Take the constant out:

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

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

Apply the Product Rule:

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

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

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

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

\frac{d}{d x} ((1 + x^{2})^{- \frac{3}{2}}) = - \frac{3 x}{( 1 + x ^{2} ) ^{\frac{5}{2}}}

= - (1 \cdot (1 + x^{2})^{- \frac{3}{2}} + (- \frac{3 x}{( 1 + x ^{2} ) ^{\frac{5}{2}}}) x)

Simplify

- (1 \cdot (1 + x^{2})^{- \frac{3}{2}} + (- \frac{3 x}{( 1 + x ^{2} ) ^{\frac{5}{2}}}) x) : - \frac{- 2 x ^{2} + 1}{( x ^{2} + 1 ) ^{\frac{5}{2}}}

= - \frac{- 2 x ^{2} + 1}{( x ^{2} + 1 ) ^{\frac{5}{2}}}

What is the derivative of-x(1+x^2^{-3/2}) ?
The derivative of-x(1+x^2^{-3/2}) is -(-2x^2+1)/((x^2+1)^{5/2)}
What is the first derivative of-x(1+x^2^{-3/2}) ?
The first derivative of-x(1+x^2^{-3/2}) is -(-2x^2+1)/((x^2+1)^{5/2)}