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Derivative of:
Derivative of x^sqrt(x)
Derivative of log(x+2)
Derivative of ln1
Derivative of e^(sin(x)-2*x^3)
Identical expressions
(two +3x)(x- one)^(one / two)
(2 plus 3x)(x minus 1) to the power of (1 divide by 2)
(two plus 3x)(x minus one) to the power of (one divide by two)
(2+3x)(x-1)(1/2)
2+3xx-11/2
2+3xx-1^1/2
(2+3x)(x-1)^(1 divide by 2)
Similar expressions
(2+3x)(x+1)^(1/2)
(2-3x)(x-1)^(1/2)

The derivative of the function/ (2+3x)(x-1)^(1/2)

Derivative of (2+3x)(x-1)^(1/2)

The solution

You have entered [src]

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

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

(2 + 3*x)*sqrt(x - 1)

Detail solution

Apply the product rule:

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

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

$\frac{9 x - 4}{2 \sqrt{x - 1}}$

The answer is:

$\frac{9 x - 4}{2 \sqrt{x - 1}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /     2 + 3*x\
3*|-6 + -------|
  \      -1 + x/
----------------
           3/2  
 8*(-1 + x)

$$\frac{3 \left(-6 + \frac{3 x + 2}{x - 1}\right)}{8 \left(x - 1\right)^{\frac{3}{2}}}$$

Simplify