Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of x*e^(-2*x+1)
Derivative of (x+1)ln(x+1)
Derivative of tan(3x-1)
Derivative of sqrt(x^2+x+3)
Graphing y =:
sqrt(x^2+x+3)
Identical expressions
sqrt(x^ two +x+ three)
square root of (x squared plus x plus 3)
square root of (x to the power of two plus x plus three)
√(x^2+x+3)
sqrt(x2+x+3)
sqrtx2+x+3
sqrt(x²+x+3)
sqrt(x to the power of 2+x+3)
sqrtx^2+x+3
Similar expressions
sqrt(x^2-x+3)
sqrt(x^2+x-3)

The derivative of the function/ sqrt(x^2+x+3)

Derivative of sqrt(x^2+x+3)

The solution

You have entered [src]

   ____________
  /  2         
\/  x  + x + 3

$$\sqrt{\left(x^{2} + x\right) + 3}$$

sqrt(x^2 + x + 3)

Detail solution

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

$\frac{2 x + 1}{2 \sqrt{\left(x^{2} + x\right) + 3}}$
Now simplify:

$\frac{x + \frac{1}{2}}{\sqrt{x^{2} + x + 3}}$

The answer is:

$\frac{x + \frac{1}{2}}{\sqrt{x^{2} + x + 3}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    1/2 + x    
---------------
   ____________
  /  2         
\/  x  + x + 3

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

            /              2\
            |     (1 + 2*x) |
3*(1 + 2*x)*|-4 + ----------|
            |              2|
            \     3 + x + x /
-----------------------------
                    3/2      
        /         2\         
      8*\3 + x + x /

$$\frac{3 \left(2 x + 1\right) \left(\frac{\left(2 x + 1\right)^{2}}{x^{2} + x + 3} - 4\right)}{8 \left(x^{2} + x + 3\right)^{\frac{3}{2}}}$$

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of sqrt(x^2+x+3)

The graph:

Piecewise:

The solution