Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of (-2)/x^3
Derivative of 1/3*x^3-4*x
Derivative of f(x)=2x-3
Derivative of y=x(x-1)
Identical expressions
nine *sqrt(x)/(sqrt(x)+ one)
9 multiply by square root of (x) divide by ( square root of (x) plus 1)
nine multiply by square root of (x) divide by ( square root of (x) plus one)
9*√(x)/(√(x)+1)
9sqrt(x)/(sqrt(x)+1)
9sqrtx/sqrtx+1
9*sqrt(x) divide by (sqrt(x)+1)
Similar expressions
9*sqrt(x)/(sqrt(x)-1)

The derivative of the function/ 9*sqrt(x)/(sqrt(x)+1)

Derivative of 9*sqrt(x)/(sqrt(x)+1)

The solution

You have entered [src]

     ___ 
 9*\/ x  
---------
  ___    
\/ x  + 1

$$\frac{9 \sqrt{x}}{\sqrt{x} + 1}$$

(9*sqrt(x))/(sqrt(x) + 1)

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = 9 \sqrt{x}$ and $g{\left(x \right)} = \sqrt{x} + 1$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
  So, the result is: $\frac{9}{2 \sqrt{x}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $\sqrt{x} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
  The result is: $\frac{1}{2 \sqrt{x}}$
Now plug in to the quotient rule:

$\frac{- \frac{9}{2} + \frac{9 \left(\sqrt{x} + 1\right)}{2 \sqrt{x}}}{\left(\sqrt{x} + 1\right)^{2}}$
Now simplify:

$\frac{9}{2 \sqrt{x} \left(\sqrt{x} + 1\right)^{2}}$

The answer is:

$\frac{9}{2 \sqrt{x} \left(\sqrt{x} + 1\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        9                   9         
- -------------- + -------------------
               2       ___ /  ___    \
    /  ___    \    2*\/ x *\\/ x  + 1/
  2*\\/ x  + 1/

$$- \frac{9}{2 \left(\sqrt{x} + 1\right)^{2}} + \frac{9}{2 \sqrt{x} \left(\sqrt{x} + 1\right)}$$

Simplify

The second derivative [src]

  /                           ___ / 1           2      \\
  |                         \/ x *|---- + -------------||
  |                               | 3/2     /      ___\||
  |   1           2               \x      x*\1 + \/ x //|
9*|- ---- - ------------- + ----------------------------|
  |   3/2     /      ___\                  ___          |
  \  x      x*\1 + \/ x /            1 + \/ x           /
---------------------------------------------------------
                        /      ___\                      
                      4*\1 + \/ x /

$$\frac{9 \left(\frac{\sqrt{x} \left(\frac{2}{x \left(\sqrt{x} + 1\right)} + \frac{1}{x^{\frac{3}{2}}}\right)}{\sqrt{x} + 1} - \frac{2}{x \left(\sqrt{x} + 1\right)} - \frac{1}{x^{\frac{3}{2}}}\right)}{4 \left(\sqrt{x} + 1\right)}$$

Simplify

The third derivative [src]

   /                                                 ___ / 1           2                  2        \\
   |                         1           2         \/ x *|---- + -------------- + -----------------||
   |                        ---- + -------------         | 5/2    2 /      ___\                   2||
   |                         3/2     /      ___\         |x      x *\1 + \/ x /    3/2 /      ___\ ||
   | 1           1          x      x*\1 + \/ x /         \                        x   *\1 + \/ x / /|
27*|---- + -------------- + -------------------- - -------------------------------------------------|
   | 5/2    2 /      ___\      ___ /      ___\                               ___                    |
   \x      x *\1 + \/ x /    \/ x *\1 + \/ x /                         1 + \/ x                     /
-----------------------------------------------------------------------------------------------------
                                              /      ___\                                            
                                            8*\1 + \/ x /

$$\frac{27 \left(- \frac{\sqrt{x} \left(\frac{2}{x^{2} \left(\sqrt{x} + 1\right)} + \frac{2}{x^{\frac{3}{2}} \left(\sqrt{x} + 1\right)^{2}} + \frac{1}{x^{\frac{5}{2}}}\right)}{\sqrt{x} + 1} + \frac{1}{x^{2} \left(\sqrt{x} + 1\right)} + \frac{\frac{2}{x \left(\sqrt{x} + 1\right)} + \frac{1}{x^{\frac{3}{2}}}}{\sqrt{x} \left(\sqrt{x} + 1\right)} + \frac{1}{x^{\frac{5}{2}}}\right)}{8 \left(\sqrt{x} + 1\right)}$$

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of 9*sqrt(x)/(sqrt(x)+1)

The graph:

Piecewise:

The solution