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Derivative of:
Derivative of e^-1
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Identical expressions
(sin3x)/cos(3x)^ five
( sinus of 3x) divide by co sinus of e of (3x) to the power of 5
( sinus of 3x) divide by co sinus of e of (3x) to the power of five
(sin3x)/cos(3x)5
sin3x/cos3x5
(sin3x)/cos(3x)⁵
sin3x/cos3x^5
(sin3x) divide by cos(3x)^5

The derivative of the function/ (sin3x)/cos(3x)^5

Derivative of (sin3x)/cos(3x)^5

The solution

You have entered [src]

 sin(3*x)
---------
   5     
cos (3*x)

$$\frac{\sin{\left(3 x \right)}}{\cos^{5}{\left(3 x \right)}}$$

d / sin(3*x)\
--|---------|
dx|   5     |
  \cos (3*x)/

$$\frac{d}{d x} \frac{\sin{\left(3 x \right)}}{\cos^{5}{\left(3 x \right)}}$$

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sin{\left(3 x \right)}$ and $g{\left(x \right)} = \cos^{5}{\left(3 x \right)}$ .

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: $u^{5}$ goes to $5 u^{4}$
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:
  
  $- 15 \sin{\left(3 x \right)} \cos^{4}{\left(3 x \right)}$
Now plug in to the quotient rule:

$\frac{15 \sin^{2}{\left(3 x \right)} \cos^{4}{\left(3 x \right)} + 3 \cos^{6}{\left(3 x \right)}}{\cos^{10}{\left(3 x \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                   2     
3*cos(3*x)   15*sin (3*x)
---------- + ------------
   5             6       
cos (3*x)     cos (3*x)

$$\frac{15 \sin^{2}{\left(3 x \right)}}{\cos^{6}{\left(3 x \right)}} + \frac{3 \cos{\left(3 x \right)}}{\cos^{5}{\left(3 x \right)}}$$

Simplify

The second derivative [src]

  /           2     \         
  |     30*sin (3*x)|         
9*|14 + ------------|*sin(3*x)
  |         2       |         
  \      cos (3*x)  /         
------------------------------
             5                
          cos (3*x)

$$\frac{9 \cdot \left(\frac{30 \sin^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} + 14\right) \sin{\left(3 x \right)}}{\cos^{5}{\left(3 x \right)}}$$

Simplify

The third derivative [src]

   /                                /           2     \\
   |                         2      |     42*sin (3*x)||
   |                    5*sin (3*x)*|17 + ------------||
   |           2                    |         2       ||
   |     75*sin (3*x)               \      cos (3*x)  /|
27*|14 + ------------ + -------------------------------|
   |         2                        2                |
   \      cos (3*x)                cos (3*x)           /
--------------------------------------------------------
                          4                             
                       cos (3*x)

$$\frac{27 \cdot \left(\frac{5 \cdot \left(\frac{42 \sin^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} + 17\right) \sin^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} + \frac{75 \sin^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} + 14\right)}{\cos^{4}{\left(3 x \right)}}$$

Simplify

The graph

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Derivative of (sin3x)/cos(3x)^5

The graph:

Piecewise:

The solution