What is the derivative of sqrt(x+\sqrt{x+1)} ?

The derivative of sqrt(x+\sqrt{x+1)} is (2sqrt(x+1)+1)/(4sqrt(x+1)\sqrt{x+\sqrt{x+1)}}

What is the first derivative of sqrt(x+\sqrt{x+1)} ?

The first derivative of sqrt(x+\sqrt{x+1)} is (2sqrt(x+1)+1)/(4sqrt(x+1)\sqrt{x+\sqrt{x+1)}}

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

\frac{2 x + 1 + 1}{4 x + 1 x + x + 1}

Solution steps

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

Apply the chain rule:

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

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

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

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

Simplify

\frac{1}{2 x + x + 1} (1 + \frac{1}{2 x + 1}) : \frac{2 x + 1 + 1}{4 x + 1 x + x + 1}

= \frac{2 x + 1 + 1}{4 x + 1 x + x + 1}

What is the derivative of sqrt(x+\sqrt{x+1)} ?
The derivative of sqrt(x+\sqrt{x+1)} is (2sqrt(x+1)+1)/(4sqrt(x+1)\sqrt{x+\sqrt{x+1)}}
What is the first derivative of sqrt(x+\sqrt{x+1)} ?
The first derivative of sqrt(x+\sqrt{x+1)} is (2sqrt(x+1)+1)/(4sqrt(x+1)\sqrt{x+\sqrt{x+1)}}