Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of sin(x)^2
Derivative of x^-6
Derivative of ex^2
Derivative of e^sin(x)*cos(x)
Identical expressions
two +x*sqrt(three -x)
2 plus x multiply by square root of (3 minus x)
two plus x multiply by square root of (three minus x)
2+x*√(3-x)
2+xsqrt(3-x)
2+xsqrt3-x
Similar expressions
2+x*sqrt(3+x)
2-x*sqrt(3-x)

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

Derivative of 2+x*sqrt(3-x)

The solution

You have entered [src]

        _______
2 + x*\/ 3 - x

$$x \sqrt{3 - x} + 2$$

2 + x*sqrt(3 - x)

Detail solution

Differentiate $x \sqrt{3 - x} + 2$ term by term:
1. The derivative of the constant $2$ is zero.
2. 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 $1$
  $g{\left(x \right)} = \sqrt{3 - x}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  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 $\frac{d}{d x} \left(3 - x\right)$ :
    1. Differentiate $3 - x$ term by term:
      1. The derivative of the constant $3$ is zero.
      2. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $-1$
      The result is: $-1$
    The result of the chain rule is:
    
    $- \frac{1}{2 \sqrt{3 - x}}$
  The result is: $- \frac{x}{2 \sqrt{3 - x}} + \sqrt{3 - x}$
The result is: $- \frac{x}{2 \sqrt{3 - x}} + \sqrt{3 - x}$
Now simplify:

$\frac{3 \left(2 - x\right)}{2 \sqrt{3 - x}}$

The answer is:

$\frac{3 \left(2 - x\right)}{2 \sqrt{3 - x}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  _______        x     
\/ 3 - x  - -----------
                _______
            2*\/ 3 - x

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify