Mister Exam

Other calculators

The derivative of the function/ sqrt(5x^3+4x-3)

Derivative of sqrt(5x^3+4x-3)

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   ________________
  /    3           
\/  5*x  + 4*x - 3
$$\sqrt{\left(5 x^{3} + 4 x\right) - 3}$$ 
sqrt(5*x^3 + 4*x - 3)
Detail solution

  1. Let .

  2. Apply the power rule: goes to

  3. Then, apply the chain rule. Multiply by :

    1. Differentiate term by term:

      1. Differentiate 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: goes to

          So, the result is:

        2. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: goes to

          So, the result is:

        The result is:

      2. The derivative of the constant is zero.

      The result is:

    The result of the chain rule is:

  4. Now simplify:

The answer is:

The graph
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
             2     
         15*x      
     2 + -----     
           2       
-------------------
   ________________
  /    3           
\/  5*x  + 4*x - 3
$$\frac{\frac{15 x^{2}}{2} + 2}{\sqrt{\left(5 x^{3} + 4 x\right) - 3}}$$
Simplify
The second derivative [src] 
                      2   
           /        2\    
           \4 + 15*x /    
15*x - -------------------
         /              3\
       4*\-3 + 4*x + 5*x /
--------------------------
      _________________   
     /               3    
   \/  -3 + 4*x + 5*x
$$\frac{15 x - \frac{\left(15 x^{2} + 4\right)^{2}}{4 \left(5 x^{3} + 4 x - 3\right)}}{\sqrt{5 x^{3} + 4 x - 3}}$$
Simplify
The third derivative [src] 
  /                   3                          \
  |        /        2\               /        2\ |
  |        \4 + 15*x /          15*x*\4 + 15*x / |
3*|5 + -------------------- - -------------------|
  |                       2     /              3\|
  |      /              3\    2*\-3 + 4*x + 5*x /|
  \    8*\-3 + 4*x + 5*x /                       /
--------------------------------------------------
                  _________________               
                 /               3                
               \/  -3 + 4*x + 5*x
$$\frac{3 \left(- \frac{15 x \left(15 x^{2} + 4\right)}{2 \left(5 x^{3} + 4 x - 3\right)} + \frac{\left(15 x^{2} + 4\right)^{3}}{8 \left(5 x^{3} + 4 x - 3\right)^{2}} + 5\right)}{\sqrt{5 x^{3} + 4 x - 3}}$$
Simplify
3-я производная [src] 
  /                   3                          \
  |        /        2\               /        2\ |
  |        \4 + 15*x /          15*x*\4 + 15*x / |
3*|5 + -------------------- - -------------------|
  |                       2     /              3\|
  |      /              3\    2*\-3 + 4*x + 5*x /|
  \    8*\-3 + 4*x + 5*x /                       /
--------------------------------------------------
                  _________________               
                 /               3                
               \/  -3 + 4*x + 5*x
$$\frac{3 \left(- \frac{15 x \left(15 x^{2} + 4\right)}{2 \left(5 x^{3} + 4 x - 3\right)} + \frac{\left(15 x^{2} + 4\right)^{3}}{8 \left(5 x^{3} + 4 x - 3\right)^{2}} + 5\right)}{\sqrt{5 x^{3} + 4 x - 3}}$$
Simplify
    © Mister Exam – Calculator