Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of atan(3*x)
Derivative of 7/x^2
Derivative of (5*x-4)*(2*x^4-7*x+1)
Derivative of 2*x^3-3*x^2+6*x+1
Identical expressions
sqrt(x^ three + two)
square root of (x cubed plus 2)
square root of (x to the power of three plus two)
√(x^3+2)
sqrt(x3+2)
sqrtx3+2
sqrt(x³+2)
sqrt(x to the power of 3+2)
sqrtx^3+2
Similar expressions
sqrt(x^3-2)

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

Derivative of sqrt(x^3+2)

The solution

You have entered [src]

   ________
  /  3     
\/  x  + 2

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

  /   ________\
d |  /  3     |
--\\/  x  + 2 /
dx

$$\frac{d}{d x} \sqrt{x^{3} + 2}$$

Transform

Detail solution

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

$\frac{3 x^{2}}{2 \sqrt{x^{3} + 2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        2    
     3*x     
-------------
     ________
    /  3     
2*\/  x  + 2

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

Simplify

The second derivative [src]

    /          3   \
    |       3*x    |
3*x*|1 - ----------|
    |      /     3\|
    \    4*\2 + x //
--------------------
       ________     
      /      3      
    \/  2 + x

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

Simplify

The third derivative [src]

  /          3             6   \
  |       9*x          27*x    |
3*|1 - ---------- + -----------|
  |      /     3\             2|
  |    2*\2 + x /     /     3\ |
  \                 8*\2 + x / /
--------------------------------
             ________           
            /      3            
          \/  2 + x

$$\frac{3 \cdot \left(\frac{27 x^{6}}{8 \left(x^{3} + 2\right)^{2}} - \frac{9 x^{3}}{2 \left(x^{3} + 2\right)} + 1\right)}{\sqrt{x^{3} + 2}}$$

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of sqrt(x^3+2)

The graph:

Piecewise:

The solution