What is the derivative of ((sqrt(x))/3+9)^{3/2} ?

The derivative of ((sqrt(x))/3+9)^{3/2} is ((sqrt(x)+27)^{1/2})/(4*3^{1/2)x^{1/2}}

What is the first derivative of ((sqrt(x))/3+9)^{3/2} ?

The first derivative of ((sqrt(x))/3+9)^{3/2} is ((sqrt(x)+27)^{1/2})/(4*3^{1/2)x^{1/2}}

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derivative of

(\frac{x}{3} + 9)^{\frac{3}{2}}

\frac{( x + 27 ) ^{\frac{1}{2}}}{4 \cdot 3 ^{\frac{1}{2}} x ^{\frac{1}{2}}}

Solution steps

\frac{d}{d x} ((\frac{x}{3} + 9)^{\frac{3}{2}})

Simplify

(\frac{x}{3} + 9)^{\frac{3}{2}} : (\frac{x + 27}{3})^{\frac{3}{2}}

= \frac{d}{d x} ((\frac{x + 27}{3})^{\frac{3}{2}})

Apply the chain rule:

\frac{3 ( \frac{x + 27}{3} ) ^{\frac{1}{2}}}{2} \frac{d}{d x} (\frac{x + 27}{3})

= \frac{3 ( \frac{x + 27}{3} ) ^{\frac{1}{2}}}{2} \frac{d}{d x} (\frac{x + 27}{3})

\frac{d}{d x} (\frac{x + 27}{3}) = \frac{1}{6 x ^{\frac{1}{2}}}

= \frac{3 ( \frac{x + 27}{3} ) ^{\frac{1}{2}}}{2} \cdot \frac{1}{6 x ^{\frac{1}{2}}}

Simplify

\frac{3 ( \frac{x + 27}{3} ) ^{\frac{1}{2}}}{2} \cdot \frac{1}{6 x ^{\frac{1}{2}}} : \frac{( x + 27 ) ^{\frac{1}{2}}}{4 \cdot 3 ^{\frac{1}{2}} x ^{\frac{1}{2}}}

= \frac{( x + 27 ) ^{\frac{1}{2}}}{4 \cdot 3 ^{\frac{1}{2}} x ^{\frac{1}{2}}}

\int_{0}^{- 1} \frac{x ^{3} + 8}{x + 2} d x

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

\frac{d}{d x} (ln e^{1 + x^{2}} + 3 x^{4} + cot^{2} (π x))

(\frac{1}{1 + e ^{- x}})^{^{'}}

6 x^{\frac{3}{5}}

What is the derivative of ((sqrt(x))/3+9)^{3/2} ?
The derivative of ((sqrt(x))/3+9)^{3/2} is ((sqrt(x)+27)^{1/2})/(4*3^{1/2)x^{1/2}}
What is the first derivative of ((sqrt(x))/3+9)^{3/2} ?
The first derivative of ((sqrt(x))/3+9)^{3/2} is ((sqrt(x)+27)^{1/2})/(4*3^{1/2)x^{1/2}}