What is the derivative of \sqrt[5]{sqrt(5x^{3/2)}} ?

The derivative of \sqrt[5]{sqrt(5x^{3/2)}} is 3/(4*5^{9/10)x^{17/20}}

What is the first derivative of \sqrt[5]{sqrt(5x^{3/2)}} ?

The first derivative of \sqrt[5]{sqrt(5x^{3/2)}} is 3/(4*5^{9/10)x^{17/20}}

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\frac{3}{4 \cdot 5 ^{\frac{9}{10}} x ^{\frac{17}{20}}}

Solution steps

Apply the chain rule:

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

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

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

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

Simplify

\frac{1}{5 ( 5 x ^{\frac{3}{2}} ) ^{\frac{4}{5}}} \cdot \frac{3 \cdot 5 ^{\frac{1}{2}} x ^{\frac{1}{2}}}{4 ( x ^{\frac{3}{2}} ) ^{\frac{1}{2}}} : \frac{3}{4 \cdot 5 ^{\frac{9}{10}} x ^{\frac{17}{20}}}

= \frac{3}{4 \cdot 5 ^{\frac{9}{10}} x ^{\frac{17}{20}}}

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

\int 7 tan^{3} (x) sec (x) d x

\frac{\partial}{\partial y} (1 + x^{3} + y^{4})

\frac{d x}{d t} = 0.25 - \frac{1}{20} x, x (0) = 0.5

y^{^{''}} + 25 y = 3 sec^{3} (5)

What is the derivative of \sqrt[5]{sqrt(5x^{3/2)}} ?
The derivative of \sqrt[5]{sqrt(5x^{3/2)}} is 3/(4*5^{9/10)x^{17/20}}
What is the first derivative of \sqrt[5]{sqrt(5x^{3/2)}} ?
The first derivative of \sqrt[5]{sqrt(5x^{3/2)}} is 3/(4*5^{9/10)x^{17/20}}