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

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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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