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

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

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

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

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derivative of

\frac{1}{3} (y^{2} + 2)^{\frac{3}{2}}

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

Solution steps

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

Take the constant out:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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