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

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

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

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

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derivative of

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

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

Solution steps

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

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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

\frac{( 4 x ^{3} - 5 x - 3 )}{( 2 x )}

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

\frac{d}{d q} ((1 + csc (q)) cos (q))

\frac{sin ( x )}{2}

(\frac{x - 1}{x})^{3}

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