Mister Exam

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The derivative of the function/ (3x-5)√x

Derivative of (3x-5)√x

The solution

You have entered [src]

            ___
(3*x - 5)*\/ x

$$\sqrt{x} \left(3 x - 5\right)$$

(3*x - 5)*sqrt(x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = 3 x - 5$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $3 x - 5$ term by term:
  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: $3$
  2. The derivative of the constant $-5$ is zero.
  The result is: $3$
$g{\left(x \right)} = \sqrt{x}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
The result is: $3 \sqrt{x} + \frac{3 x - 5}{2 \sqrt{x}}$
Now simplify:

$\frac{9 x - 5}{2 \sqrt{x}}$

The answer is:

$\frac{9 x - 5}{2 \sqrt{x}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    ___   3*x - 5
3*\/ x  + -------
              ___
          2*\/ x

$$3 \sqrt{x} + \frac{3 x - 5}{2 \sqrt{x}}$$

Simplify

The second derivative [src]

    -5 + 3*x
3 - --------
      4*x   
------------
     ___    
   \/ x

$$\frac{3 - \frac{3 x - 5}{4 x}}{\sqrt{x}}$$

Simplify

The third derivative [src]

  /     -5 + 3*x\
3*|-6 + --------|
  \        x    /
-----------------
         3/2     
      8*x

$$\frac{3 \left(-6 + \frac{3 x - 5}{x}\right)}{8 x^{\frac{3}{2}}}$$

Simplify

The graph

Derivative of (3x-5)√x