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Derivative of:
Derivative of 5*x
Derivative of x-5
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Derivative of 4x^2
Identical expressions
sqrt3x+sqrt^2x+ one /x
square root of 3x plus square root of squared x plus 1 divide by x
square root of 3x plus square root of squared x plus one divide by x
√3x+√^2x+1/x
sqrt3x+sqrt2x+1/x
sqrt3x+sqrt²x+1/x
sqrt3x+sqrt to the power of 2x+1/x
sqrt3x+sqrt^2x+1 divide by x
Similar expressions
sqrt3x+sqrt^2x-1/x
sqrt3x-sqrt^2x+1/x

The derivative of the function/ sqrt3x+sqrt^2x+1/x

Derivative of sqrt3x+sqrt^2x+1/x

The solution

You have entered [src]

               2    
  _____     ___    1
\/ 3*x  + \/ x   + -
                   x

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

sqrt(3*x) + (sqrt(x))^2 + 1/x

Detail solution

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

             ___   ___
  1    x   \/ 3 *\/ x 
- -- + - + -----------
   2   x       2*x    
  x

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

Simplify

The second derivative [src]

       ___ 
2    \/ 3  
-- - ------
 3      3/2
x    4*x

$$\frac{2}{x^{3}} - \frac{\sqrt{3}}{4 x^{\frac{3}{2}}}$$

Simplify

The third derivative [src]

  /         ___ \
  |  2    \/ 3  |
3*|- -- + ------|
  |   4      5/2|
  \  x    8*x   /

$$3 \left(- \frac{2}{x^{4}} + \frac{\sqrt{3}}{8 x^{\frac{5}{2}}}\right)$$

Simplify