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Derivative of:
Derivative of x/16
Derivative of sin9x
Derivative of (ln(x+8)-3x+2)/x
Derivative of (ln(x+8)-3(x)+2)/x
Identical expressions
sqrt(109cos^ twenty)
square root of (109 co sinus of e of squared 0)
square root of (109 co sinus of e of to the power of twenty)
√(109cos^20)
sqrt(109cos20)
sqrt109cos20
sqrt(109cos²0)
sqrt(109cos to the power of 20)
sqrt109cos^20

The derivative of the function/ sqrt(109cos^20)

Derivative of sqrt(109cos^20)

The solution

You have entered [src]

   ______________
  /        20    
\/  109*cos  (x)

$$\sqrt{109 \cos^{20}{\left(x \right)}}$$

sqrt(109*cos(x)^20)

Detail solution

Let $u = 109 \cos^{20}{\left(x \right)}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} 109 \cos^{20}{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \cos{\left(x \right)}$ .
  2. Apply the power rule: $u^{20}$ goes to $20 u^{19}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    The result of the chain rule is:
    
    $- 20 \sin{\left(x \right)} \cos^{19}{\left(x \right)}$
  So, the result is: $- 2180 \sin{\left(x \right)} \cos^{19}{\left(x \right)}$
The result of the chain rule is:

$- 10 \sqrt{109} \sin{\left(x \right)} \cos^{9}{\left(x \right)}$

The answer is:

$- 10 \sqrt{109} \sin{\left(x \right)} \cos^{9}{\left(x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      _____    10          
-10*\/ 109 *cos  (x)*sin(x)
---------------------------
           cos(x)

$$- \frac{10 \sqrt{109} \cos^{10}{\left(x \right)} \sin{\left(x \right)}}{\cos{\left(x \right)}}$$

Simplify

The second derivative [src]

     _____    8    /     2           2   \
10*\/ 109 *cos (x)*\- cos (x) + 9*sin (x)/

$$10 \sqrt{109} \left(9 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos^{8}{\left(x \right)}$$

Simplify

The third derivative [src]

     _____    7    /        2           2   \       
40*\/ 109 *cos (x)*\- 18*sin (x) + 7*cos (x)/*sin(x)

$$40 \sqrt{109} \left(- 18 \sin^{2}{\left(x \right)} + 7 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \cos^{7}{\left(x \right)}$$

Simplify