What is the derivative of (5x^2-15/(3(x^2-5)^{2/3)}) ?

The derivative of (5x^2-15/(3(x^2-5)^{2/3)}) is (10x^3-90x)/(9(x^2-5)^{5/3)}

What is the first derivative of (5x^2-15/(3(x^2-5)^{2/3)}) ?

The first derivative of (5x^2-15/(3(x^2-5)^{2/3)}) is (10x^3-90x)/(9(x^2-5)^{5/3)}

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\frac{d}{d x} (\frac{5 x ^{2} - 15}{3 ( x ^{2} - 5 ) ^{\frac{2}{3}}})

\frac{10 x ^{3} - 90 x}{9 ( x ^{2} - 5 ) ^{\frac{5}{3}}}

Solution steps

\frac{d}{d x} (\frac{5 x ^{2} - 15}{3 ( x ^{2} - 5 ) ^{\frac{2}{3}}})

Take the constant out:

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

= \frac{1}{3} \frac{d}{d x} (\frac{5 x ^{2} - 15}{( x ^{2} - 5 ) ^{\frac{2}{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} ( 5 x ^{2} - 15 ) ( x ^{2} - 5 ) ^{\frac{2}{3}} - \frac{d}{d x} ( ( x ^{2} - 5 ) ^{\frac{2}{3}} ) ( 5 x ^{2} - 15 )}{( ( x ^{2} - 5 ) ^{\frac{2}{3}} ) ^{2}}

\frac{d}{d x} (5 x^{2} - 15) = 10 x

\frac{d}{d x} ((x^{2} - 5)^{\frac{2}{3}}) = \frac{4 x}{3 ( x ^{2} - 5 ) ^{\frac{1}{3}}}

= \frac{1}{3} \cdot \frac{10 x ( x ^{2} - 5 ) ^{\frac{2}{3}} - \frac{4 x}{3 ( x ^{2} - 5 ) ^{\frac{1}{3}}} ( 5 x ^{2} - 15 )}{( ( x ^{2} - 5 ) ^{\frac{2}{3}} ) ^{2}}

Simplify

\frac{1}{3} \cdot \frac{10 x ( x ^{2} - 5 ) ^{\frac{2}{3}} - \frac{4 x}{3 ( x ^{2} - 5 ) ^{\frac{1}{3}}} ( 5 x ^{2} - 15 )}{( ( x ^{2} - 5 ) ^{\frac{2}{3}} ) ^{2}} : \frac{10 x ^{3} - 90 x}{9 ( x ^{2} - 5 ) ^{\frac{5}{3}}}

= \frac{10 x ^{3} - 90 x}{9 ( x ^{2} - 5 ) ^{\frac{5}{3}}}

What is the derivative of (5x^2-15/(3(x^2-5)^{2/3)}) ?
The derivative of (5x^2-15/(3(x^2-5)^{2/3)}) is (10x^3-90x)/(9(x^2-5)^{5/3)}
What is the first derivative of (5x^2-15/(3(x^2-5)^{2/3)}) ?
The first derivative of (5x^2-15/(3(x^2-5)^{2/3)}) is (10x^3-90x)/(9(x^2-5)^{5/3)}