Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of 2/e^x
Derivative of sec2x
Derivative of f(x)=2x+3
Derivative of e^x-7
Identical expressions
ln(sqrt(x*x+ two))
ln( square root of (x multiply by x plus 2))
ln( square root of (x multiply by x plus two))
ln(√(x*x+2))
ln(sqrt(xx+2))
lnsqrtxx+2
Similar expressions
ln(sqrt(x*x-2))

The derivative of the function/ ln(sqrt(x*x+2))

Derivative of ln(sqrt(x*x+2))

The solution

You have entered [src]

   /  _________\
log\\/ x*x + 2 /

$$\log{\left(\sqrt{x x + 2} \right)}$$

log(sqrt(x*x + 2))

Detail solution

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

$\frac{x}{x x + 2}$
Now simplify:

$\frac{x}{x^{2} + 2}$

The answer is:

$\frac{x}{x^{2} + 2}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   x   
-------
x*x + 2

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

Simplify

The second derivative [src]

        2 
     2*x  
1 - ------
         2
    2 + x 
----------
       2  
  2 + x

$$\frac{- \frac{2 x^{2}}{x^{2} + 2} + 1}{x^{2} + 2}$$

Simplify

The third derivative [src]

    /         2 \
    |      4*x  |
2*x*|-3 + ------|
    |          2|
    \     2 + x /
-----------------
            2    
    /     2\     
    \2 + x /

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

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of ln(sqrt(x*x+2))

The graph:

Piecewise:

The solution