Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of x*e^x+11
Derivative of x*cos(3*x)
Derivative of x^5-2*x
Derivative of (x-4)^3
Identical expressions
y=ln(x+√ four +x^ two)
y equally ln(x plus √4 plus x squared )
y equally ln(x plus √ four plus x to the power of two)
y=ln(x+√4+x2)
y=lnx+√4+x2
y=ln(x+√4+x²)
y=ln(x+√4+x to the power of 2)
y=lnx+√4+x^2
Similar expressions
y=ln(x-√4+x^2)
y=ln(x+√4-x^2)

The derivative of the function/ y=ln(x+√4+x^2)

Derivative of y=ln(x+√4+x^2)

The solution

You have entered [src]

   /      ___    2\
log\x + \/ 4  + x /

$$\log{\left(x^{2} + x + \sqrt{4} \right)}$$

d /   /      ___    2\\
--\log\x + \/ 4  + x //
dx

$$\frac{d}{d x} \log{\left(x^{2} + x + \sqrt{4} \right)}$$

Transform

Detail solution

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

$\frac{2 x + 1}{x^{2} + x + \sqrt{4}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   1 + 2*x    
--------------
      ___    2
x + \/ 4  + x

$$\frac{2 x + 1}{x^{2} + x + \sqrt{4}}$$

Simplify

The second derivative [src]

             2
    (1 + 2*x) 
2 - ----------
             2
    2 + x + x 
--------------
           2  
  2 + x + x

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

Simplify

The third derivative [src]

            /              2\
            |     (1 + 2*x) |
2*(1 + 2*x)*|-3 + ----------|
            |              2|
            \     2 + x + x /
-----------------------------
                    2        
        /         2\         
        \2 + x + x /

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

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of y=ln(x+√4+x^2)

The graph:

Piecewise:

The solution