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

The derivative of 1/(2(x+1^{1/2)}) is -1/(4(x+1)^{3/2)}

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

The first derivative of 1/(2(x+1^{1/2)}) is -1/(4(x+1)^{3/2)}

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

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

Solution steps

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

Take the constant out:

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

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

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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