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Derivative of:
Derivative of e^x*x^2
Derivative of 2*sqrt(x^3)
Derivative of (x-4)^3
Derivative of x^3-5x
Integral of d{x}:
cos((x^3)/3)
Identical expressions
y=cos((x^ three)/ three)
y equally co sinus of e of ((x cubed ) divide by 3)
y equally co sinus of e of ((x to the power of three) divide by three)
y=cos((x3)/3)
y=cosx3/3
y=cos((x³)/3)
y=cos((x to the power of 3)/3)
y=cosx^3/3
y=cos((x^3) divide by 3)

The derivative of the function/ y=cos((x^3)/3)

Derivative of y=cos((x^3)/3)

The solution

You have entered [src]

   / 3\
   |x |
cos|--|
   \3 /

$$\cos{\left(\frac{x^{3}}{3} \right)}$$

  /   / 3\\
d |   |x ||
--|cos|--||
dx\   \3 //

$$\frac{d}{d x} \cos{\left(\frac{x^{3}}{3} \right)}$$

Transform

Detail solution

Let $u = \frac{x^{3}}{3}$ .
The derivative of cosine is negative sine:

$\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{x^{3}}{3}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
  So, the result is: $x^{2}$
The result of the chain rule is:

$- x^{2} \sin{\left(\frac{x^{3}}{3} \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       / 3\
  2    |x |
-x *sin|--|
       \3 /

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

Simplify

The second derivative [src]

   /     / 3\         / 3\\
   |     |x |    3    |x ||
-x*|2*sin|--| + x *cos|--||
   \     \3 /         \3 //

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

Simplify

4-th derivative [src]

   /        / 3\         / 3\            / 3\\
 2 |        |x |    6    |x |       3    |x ||
x *|- 20*cos|--| + x *cos|--| + 12*x *sin|--||
   \        \3 /         \3 /            \3 //

$$x^{2} \left(x^{6} \cos{\left(\frac{x^{3}}{3} \right)} + 12 x^{3} \sin{\left(\frac{x^{3}}{3} \right)} - 20 \cos{\left(\frac{x^{3}}{3} \right)}\right)$$

Simplify

The third derivative [src]

       / 3\         / 3\           / 3\
       |x |    6    |x |      3    |x |
- 2*sin|--| + x *sin|--| - 6*x *cos|--|
       \3 /         \3 /           \3 /

$$x^{6} \sin{\left(\frac{x^{3}}{3} \right)} - 6 x^{3} \cos{\left(\frac{x^{3}}{3} \right)} - 2 \sin{\left(\frac{x^{3}}{3} \right)}$$

Simplify

The graph

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Derivative of y=cos((x^3)/3)

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The solution