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

The derivative of ((x^2-1^{3/2})/(3x^3)) is ((x^2-1)^{1/2})/(x^4)

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

The first derivative of ((x^2-1^{3/2})/(3x^3)) is ((x^2-1)^{1/2})/(x^4)

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

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

Solution steps

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

Take the constant out:

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

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

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

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

Simplify

\frac{1}{3} \cdot \frac{3 x ( x ^{2} - 1 ) ^{\frac{1}{2}} x ^{3} - 3 x ^{2} ( x ^{2} - 1 ) ^{\frac{3}{2}}}{( x ^{3} ) ^{2}} : \frac{( x ^{2} - 1 ) ^{\frac{1}{2}}}{x ^{4}}

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

What is the derivative of ((x^2-1^{3/2})/(3x^3)) ?
The derivative of ((x^2-1^{3/2})/(3x^3)) is ((x^2-1)^{1/2})/(x^4)
What is the first derivative of ((x^2-1^{3/2})/(3x^3)) ?
The first derivative of ((x^2-1^{3/2})/(3x^3)) is ((x^2-1)^{1/2})/(x^4)