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Identical expressions
(5cosx+sqrt(x)^ three)^ four
(5 co sinus of e of x plus square root of (x) cubed ) to the power of 4
(5 co sinus of e of x plus square root of (x) to the power of three) to the power of four
(5cosx+√(x)^3)^4
(5cosx+sqrt(x)3)4
5cosx+sqrtx34
(5cosx+sqrt(x)³)⁴
(5cosx+sqrt(x) to the power of 3) to the power of 4
5cosx+sqrtx^3^4
Similar expressions
(5cosx-sqrt(x)^3)^4

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

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

The solution

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                   4
/                3\ 
|             ___ | 
\5*cos(x) + \/ x  /

$$\left(\left(\sqrt{x}\right)^{3} + 5 \cos{\left(x \right)}\right)^{4}$$

  /                   4\
  |/                3\ |
d ||             ___ | |
--\\5*cos(x) + \/ x  / /
dx

$$\frac{d}{d x} \left(\left(\sqrt{x}\right)^{3} + 5 \cos{\left(x \right)}\right)^{4}$$

Transform

Detail solution

Let $u = \left(\sqrt{x}\right)^{3} + 5 \cos{\left(x \right)}$ .
Apply the power rule: $u^{4}$ goes to $4 u^{3}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\left(\sqrt{x}\right)^{3} + 5 \cos{\left(x \right)}\right)$ :
1. Differentiate $\left(\sqrt{x}\right)^{3} + 5 \cos{\left(x \right)}$ term by term:
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    So, the result is: $- 5 \sin{\left(x \right)}$
  2. Let $u = \sqrt{x}$ .
  3. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
  4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{x}$ :
    1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
    The result of the chain rule is:
    
    $\frac{3 \sqrt{x}}{2}$
  The result is: $\frac{3 \sqrt{x}}{2} - 5 \sin{\left(x \right)}$
The result of the chain rule is:

$4 \cdot \left(\frac{3 \sqrt{x}}{2} - 5 \sin{\left(x \right)}\right) \left(\left(\sqrt{x}\right)^{3} + 5 \cos{\left(x \right)}\right)^{3}$
Now simplify:

$\left(6 \sqrt{x} - 20 \sin{\left(x \right)}\right) \left(x^{\frac{3}{2}} + 5 \cos{\left(x \right)}\right)^{3}$

The answer is:

$\left(6 \sqrt{x} - 20 \sin{\left(x \right)}\right) \left(x^{\frac{3}{2}} + 5 \cos{\left(x \right)}\right)^{3}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                   3                      
/                3\  /                3/2\
|             ___ |  |             6*x   |
\5*cos(x) + \/ x  / *|-20*sin(x) + ------|
                     \               x   /

$$\left(\frac{6 x^{\frac{3}{2}}}{x} - 20 \sin{\left(x \right)}\right) \left(\left(\sqrt{x}\right)^{3} + 5 \cos{\left(x \right)}\right)^{3}$$

Simplify

The second derivative [src]

                 2 /                        2                                          \
/ 3/2           \  |  /                 ___\    / 3/2           \ /    3              \|
\x    + 5*cos(x)/ *|3*\-10*sin(x) + 3*\/ x /  - \x    + 5*cos(x)/*|- ----- + 20*cos(x)||
                   |                                              |    ___            ||
                   \                                              \  \/ x             //

$$\left(x^{\frac{3}{2}} + 5 \cos{\left(x \right)}\right)^{2} \cdot \left(3 \left(3 \sqrt{x} - 10 \sin{\left(x \right)}\right)^{2} - \left(x^{\frac{3}{2}} + 5 \cos{\left(x \right)}\right) \left(20 \cos{\left(x \right)} - \frac{3}{\sqrt{x}}\right)\right)$$

Simplify

The third derivative [src]

/ 3/2           \ /                        3                    2                                                                                        \
|x      5*cos(x)| |  /                 ___\    / 3/2           \  /   3              \     / 3/2           \ /                 ___\ /    3              \|
|---- + --------|*|6*\-10*sin(x) + 3*\/ x /  + \x    + 5*cos(x)/ *|- ---- + 40*sin(x)| - 9*\x    + 5*cos(x)/*\-10*sin(x) + 3*\/ x /*|- ----- + 20*cos(x)||
\ 2        2    / |                                               |   3/2            |                                              |    ___            ||
                  \                                               \  x               /                                              \  \/ x             //

$$\left(\frac{x^{\frac{3}{2}}}{2} + \frac{5 \cos{\left(x \right)}}{2}\right) \left(6 \left(3 \sqrt{x} - 10 \sin{\left(x \right)}\right)^{3} - 9 \cdot \left(3 \sqrt{x} - 10 \sin{\left(x \right)}\right) \left(x^{\frac{3}{2}} + 5 \cos{\left(x \right)}\right) \left(20 \cos{\left(x \right)} - \frac{3}{\sqrt{x}}\right) + \left(x^{\frac{3}{2}} + 5 \cos{\left(x \right)}\right)^{2} \cdot \left(40 \sin{\left(x \right)} - \frac{3}{x^{\frac{3}{2}}}\right)\right)$$

Simplify

The graph

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

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The solution