Mister Exam

Lang:

The derivative of the function/ sqrt(x+1)-1

Derivative of sqrt(x+1)-1

The solution

You have entered [src]

  _______    
\/ x + 1  - 1

$$\sqrt{x + 1} - 1$$

sqrt(x + 1) - 1

Detail solution

Differentiate $\sqrt{x + 1} - 1$ term by term:
1. Let $u = x + 1$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 1\right)$ :
  1. Differentiate $x + 1$ term by term:
    1. Apply the power rule: $x$ goes to $1$
    2. The derivative of the constant $1$ is zero.
    The result is: $1$
  The result of the chain rule is:
  
  $\frac{1}{2 \sqrt{x + 1}}$
4. The derivative of the constant $-1$ is zero.
The result is: $\frac{1}{2 \sqrt{x + 1}}$
Now simplify:

$\frac{1}{2 \sqrt{x + 1}}$

The answer is:

$\frac{1}{2 \sqrt{x + 1}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     1     
-----------
    _______
2*\/ x + 1

$$\frac{1}{2 \sqrt{x + 1}}$$

Simplify

The second derivative [src]

    -1      
------------
         3/2
4*(1 + x)

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

Simplify

The third derivative [src]

     3      
------------
         5/2
8*(1 + x)

$$\frac{3}{8 \left(x + 1\right)^{\frac{5}{2}}}$$

Simplify