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

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

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

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

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

x (x^{2} + 2)^{\frac{1}{2}}

Solution steps

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

Take the constant out:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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t^{4} y^{^{''}} + t^{3} y^{^{'}} = 0

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