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Identical expressions
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Similar expressions
9*sin(3*x)/(cos(3*x))^2

The derivative of the function/ -9*sin(3*x)/(cos(3*x))^2

Derivative of -9sin(3x)/(cos(3*x))^2

The solution

You have entered [src]

-9*sin(3*x)
-----------
    2      
 cos (3*x)

$$- \frac{9 \sin{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}}$$

d /-9*sin(3*x)\
--|-----------|
dx|    2      |
  \ cos (3*x) /

$$\frac{d}{d x} \left(- \frac{9 \sin{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}}\right)$$

Transform

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(3 x \right)}$ and $g{\left(x \right)} = \cos^{2}{\left(3 x \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  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:
    
    $3 \cos{\left(3 x \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = \cos{\left(3 x \right)}$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  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.
        
        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:
    
    $- 6 \sin{\left(3 x \right)} \cos{\left(3 x \right)}$
  Now plug in to the quotient rule:
  
  $\frac{6 \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)} + 3 \cos^{3}{\left(3 x \right)}}{\cos^{4}{\left(3 x \right)}}$
So, the result is: $- \frac{9 \cdot \left(6 \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)} + 3 \cos^{3}{\left(3 x \right)}\right)}{\cos^{4}{\left(3 x \right)}}$
Now simplify:

$\frac{27 \left(\cos{\left(6 x \right)} - 3\right)}{2 \cos^{3}{\left(3 x \right)}}$

The answer is:

$\frac{27 \left(\cos{\left(6 x \right)} - 3\right)}{2 \cos^{3}{\left(3 x \right)}}$

The graph

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The first derivative [src]

        2                   
  54*sin (3*x)   27*cos(3*x)
- ------------ - -----------
      3              2      
   cos (3*x)      cos (3*x)

$$- \frac{27 \cos{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} - \frac{54 \sin^{2}{\left(3 x \right)}}{\cos^{3}{\left(3 x \right)}}$$

Simplify

The second derivative [src]

    /         2     \         
    |    6*sin (3*x)|         
-81*|5 + -----------|*sin(3*x)
    |        2      |         
    \     cos (3*x) /         
------------------------------
             2                
          cos (3*x)

$$- \frac{81 \cdot \left(\frac{6 \sin^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} + 5\right) \sin{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}}$$

Simplify

The third derivative [src]

    /                                /         2     \\
    |                         2      |    3*sin (3*x)||
    |                    8*sin (3*x)*|2 + -----------||
    |           2                    |        2      ||
    |     12*sin (3*x)               \     cos (3*x) /|
243*|-5 - ------------ - -----------------------------|
    |         2                       2               |
    \      cos (3*x)               cos (3*x)          /
-------------------------------------------------------
                        cos(3*x)

$$\frac{243 \left(- \frac{8 \cdot \left(\frac{3 \sin^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} + 2\right) \sin^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} - \frac{12 \sin^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} - 5\right)}{\cos{\left(3 x \right)}}$$

Simplify

The graph

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Derivative of -9sin(3x)/(cos(3*x))^2

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The solution

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How to use it?

Identical expressions

Similar expressions

Derivative of -9*sin(3*x)/(cos(3*x))^2

The graph:

Piecewise:

The solution

Derivative of -9sin(3x)/(cos(3*x))^2