What is the derivative of (-3x-sqrt(16x))/(2x^{1/3)} ?

The derivative of (-3x-sqrt(16x))/(2x^{1/3)} is -(3x^{1/2}+1)/(3x^{5/6)}

What is the first derivative of (-3x-sqrt(16x))/(2x^{1/3)} ?

The first derivative of (-3x-sqrt(16x))/(2x^{1/3)} is -(3x^{1/2}+1)/(3x^{5/6)}

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derivative of

\frac{- 3 x - 16 x}{2 x ^{\frac{1}{3}}}

- \frac{3 x ^{\frac{1}{2}} + 1}{3 x ^{\frac{5}{6}}}

Solution steps

\frac{d}{d x} (\frac{- 3 x - 16 x}{2 x ^{\frac{1}{3}}})

Simplify

\frac{- 3 x - 16 x}{2 x ^{\frac{1}{3}}} : - \frac{x ^{\frac{1}{6}} ( 3 x ^{\frac{1}{2}} + 4 )}{2}

= \frac{d}{d x} - \frac{x ^{\frac{1}{6}} ( 3 x ^{\frac{1}{2}} + 4 )}{2}

Take the constant out:

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

= - \frac{1}{2} \frac{d}{d x} (x^{\frac{1}{6}} (3 x^{\frac{1}{2}} + 4))

Apply the Product Rule:

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

f = x^{\frac{1}{6}}, g = 3 x^{\frac{1}{2}} + 4

= - \frac{1}{2} (\frac{d}{d x} (x^{\frac{1}{6}}) (3 x^{\frac{1}{2}} + 4) + \frac{d}{d x} (3 x^{\frac{1}{2}} + 4) x^{\frac{1}{6}})

\frac{d}{d x} (x^{\frac{1}{6}}) = \frac{1}{6 x ^{\frac{5}{6}}}

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

= - \frac{1}{2} (\frac{1}{6 x ^{\frac{5}{6}}} (3 x^{\frac{1}{2}} + 4) + \frac{3}{2 x ^{\frac{1}{2}}} x^{\frac{1}{6}})

Simplify

- \frac{1}{2} (\frac{1}{6 x ^{\frac{5}{6}}} (3 x^{\frac{1}{2}} + 4) + \frac{3}{2 x ^{\frac{1}{2}}} x^{\frac{1}{6}}) : - \frac{3 x ^{\frac{1}{2}} + 1}{3 x ^{\frac{5}{6}}}

= - \frac{3 x ^{\frac{1}{2}} + 1}{3 x ^{\frac{5}{6}}}

y^{^{''}} + 20 y^{^{'}} + 200 y = 0

y^{^{''}} + 8 y^{^{'}} + 13 y = 0

x \frac{d y}{d x} - y = x^{2} y^{2}

\frac{1}{( x ^{2} + 25 ) ^{- 3}}

y^{^{'}} + 20 y = 24

What is the derivative of (-3x-sqrt(16x))/(2x^{1/3)} ?
The derivative of (-3x-sqrt(16x))/(2x^{1/3)} is -(3x^{1/2}+1)/(3x^{5/6)}
What is the first derivative of (-3x-sqrt(16x))/(2x^{1/3)} ?
The first derivative of (-3x-sqrt(16x))/(2x^{1/3)} is -(3x^{1/2}+1)/(3x^{5/6)}