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(18*sin(3x)) divide by (cos(3x))^3

The derivative of the function/ (18*sin(3x))/(cos(3x))^3

Derivative of (18*sin(3x))/(cos(3x))^3

The solution

You have entered [src]

18*sin(3*x)
-----------
    3      
 cos (3*x)

$$\frac{18 \sin{\left(3 x \right)}}{\cos^{3}{\left(3 x \right)}}$$

(18*sin(3*x))/cos(3*x)^3

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = 18 \sin{\left(3 x \right)}$ and $g{\left(x \right)} = \cos^{3}{\left(3 x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = 3 x$ .
  2. The derivative of sine is cosine:
    
    $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $3$
    The result of the chain rule is:
    
    $3 \cos{\left(3 x \right)}$
  So, the result is: $54 \cos{\left(3 x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \cos{\left(3 x \right)}$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(3 x \right)}$ :
  1. Let $u = 3 x$ .
  2. The derivative of cosine is negative sine:
    
    $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $3$
    The result of the chain rule is:
    
    $- 3 \sin{\left(3 x \right)}$
  The result of the chain rule is:
  
  $- 9 \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}$
Now plug in to the quotient rule:

$\frac{162 \sin^{2}{\left(3 x \right)} \cos^{2}{\left(3 x \right)} + 54 \cos^{4}{\left(3 x \right)}}{\cos^{6}{\left(3 x \right)}}$
Now simplify:

$\frac{54 \left(2 - \cos{\left(6 x \right)}\right)}{\cos^{4}{\left(3 x \right)}}$

The answer is:

$\frac{54 \left(2 - \cos{\left(6 x \right)}\right)}{\cos^{4}{\left(3 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                     2     
54*cos(3*x)   162*sin (3*x)
----------- + -------------
    3              4       
 cos (3*x)      cos (3*x)

$$\frac{162 \sin^{2}{\left(3 x \right)}}{\cos^{4}{\left(3 x \right)}} + \frac{54 \cos{\left(3 x \right)}}{\cos^{3}{\left(3 x \right)}}$$

Simplify

The second derivative [src]

    /          2     \         
    |    12*sin (3*x)|         
162*|8 + ------------|*sin(3*x)
    |        2       |         
    \     cos (3*x)  /         
-------------------------------
              3                
           cos (3*x)

$$\frac{162 \left(\frac{12 \sin^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} + 8\right) \sin{\left(3 x \right)}}{\cos^{3}{\left(3 x \right)}}$$

Simplify

The third derivative [src]

    /                               /           2     \\
    |                        2      |     20*sin (3*x)||
    |                   3*sin (3*x)*|11 + ------------||
    |          2                    |         2       ||
    |    27*sin (3*x)               \      cos (3*x)  /|
486*|8 + ------------ + -------------------------------|
    |        2                        2                |
    \     cos (3*x)                cos (3*x)           /
--------------------------------------------------------
                          2                             
                       cos (3*x)

$$\frac{486 \left(\frac{3 \left(\frac{20 \sin^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} + 11\right) \sin^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} + \frac{27 \sin^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} + 8\right)}{\cos^{2}{\left(3 x \right)}}$$

Simplify

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Derivative of (18*sin(3x))/(cos(3x))^3

The graph:

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The solution