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The derivative of the function/ tan(3*x)*sin(x/2)^3

Derivative of tan(3x)sin(x/2)^3

The solution

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            3/x\
tan(3*x)*sin |-|
             \2/

$$\sin^{3}{\left(\frac{x}{2} \right)} \tan{\left(3 x \right)}$$

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

$$\frac{d}{d x} \sin^{3}{\left(\frac{x}{2} \right)} \tan{\left(3 x \right)}$$

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \tan{\left(3 x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(3 x \right)} = \frac{\sin{\left(3 x \right)}}{\cos{\left(3 x \right)}}$
2. 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 $g{\left(x \right)} = \cos{\left(3 x \right)}$ .
  
  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 $u = 3 x$ .
  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:
    
    $- 3 \sin{\left(3 x \right)}$
  Now plug in to the quotient rule:
  
  $\frac{3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}}$
$g{\left(x \right)} = \sin^{3}{\left(\frac{x}{2} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \sin{\left(\frac{x}{2} \right)}$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(\frac{x}{2} \right)}$ :
  1. Let $u = \frac{x}{2}$ .
  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 $\frac{d}{d x} \frac{x}{2}$ :
    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: $\frac{1}{2}$
    The result of the chain rule is:
    
    $\frac{\cos{\left(\frac{x}{2} \right)}}{2}$
  The result of the chain rule is:
  
  $\frac{3 \sin^{2}{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}}{2}$
The result is: $\frac{\left(3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}\right) \sin^{3}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(3 x \right)}} + \frac{3 \sin^{2}{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} \tan{\left(3 x \right)}}{2}$
Now simplify:

$\frac{3 \cdot \left(2 \sin{\left(\frac{x}{2} \right)} + \frac{\sin{\left(\frac{11 x}{2} \right)}}{4} + \frac{\sin{\left(\frac{13 x}{2} \right)}}{4}\right) \sin^{2}{\left(\frac{x}{2} \right)}}{2 \cos^{2}{\left(3 x \right)}}$

The answer is:

$\frac{3 \cdot \left(2 \sin{\left(\frac{x}{2} \right)} + \frac{\sin{\left(\frac{11 x}{2} \right)}}{4} + \frac{\sin{\left(\frac{13 x}{2} \right)}}{4}\right) \sin^{2}{\left(\frac{x}{2} \right)}}{2 \cos^{2}{\left(3 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                                 2/x\    /x\         
                            3*sin |-|*cos|-|*tan(3*x)
   3/x\ /         2     \         \2/    \2/         
sin |-|*\3 + 3*tan (3*x)/ + -------------------------
    \2/                                 2

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

Simplify

The second derivative [src]

  /  /   2/x\        2/x\\                                                                                \       
  |  |sin |-| - 2*cos |-||*tan(3*x)                                                                       |       
  |  \    \2/         \2//              /       2     \    /x\    /x\        2/x\ /       2     \         |    /x\
3*|- ------------------------------ + 3*\1 + tan (3*x)/*cos|-|*sin|-| + 6*sin |-|*\1 + tan (3*x)/*tan(3*x)|*sin|-|
  \                4                                       \2/    \2/         \2/                         /    \2/

$$3 \left(- \frac{\left(\sin^{2}{\left(\frac{x}{2} \right)} - 2 \cos^{2}{\left(\frac{x}{2} \right)}\right) \tan{\left(3 x \right)}}{4} + 6 \left(\tan^{2}{\left(3 x \right)} + 1\right) \sin^{2}{\left(\frac{x}{2} \right)} \tan{\left(3 x \right)} + 3 \left(\tan^{2}{\left(3 x \right)} + 1\right) \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}\right) \sin{\left(\frac{x}{2} \right)}$$

Simplify

The third derivative [src]

  /                                                 /       2     \ /   2/x\        2/x\\    /x\   /       2/x\        2/x\\    /x\                                                      \
  |                                               9*\1 + tan (3*x)/*|sin |-| - 2*cos |-||*sin|-|   |- 2*cos |-| + 7*sin |-||*cos|-|*tan(3*x)                                             |
  |      3/x\ /       2     \ /         2     \                     \    \2/         \2//    \2/   \        \2/         \2//    \2/                  2/x\ /       2     \    /x\         |
3*|18*sin |-|*\1 + tan (3*x)/*\1 + 3*tan (3*x)/ - ---------------------------------------------- - ----------------------------------------- + 27*sin |-|*\1 + tan (3*x)/*cos|-|*tan(3*x)|
  \       \2/                                                           4                                              8                              \2/                    \2/         /

$$3 \left(- \frac{9 \left(\sin^{2}{\left(\frac{x}{2} \right)} - 2 \cos^{2}{\left(\frac{x}{2} \right)}\right) \left(\tan^{2}{\left(3 x \right)} + 1\right) \sin{\left(\frac{x}{2} \right)}}{4} - \frac{\left(7 \sin^{2}{\left(\frac{x}{2} \right)} - 2 \cos^{2}{\left(\frac{x}{2} \right)}\right) \cos{\left(\frac{x}{2} \right)} \tan{\left(3 x \right)}}{8} + 18 \left(\tan^{2}{\left(3 x \right)} + 1\right) \left(3 \tan^{2}{\left(3 x \right)} + 1\right) \sin^{3}{\left(\frac{x}{2} \right)} + 27 \left(\tan^{2}{\left(3 x \right)} + 1\right) \sin^{2}{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} \tan{\left(3 x \right)}\right)$$

Simplify

The graph

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Identical expressions

Derivative of tan(3x)sin(x/2)^3

The graph:

Piecewise:

The solution

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How to use it?

Identical expressions

Derivative of tan(3*x)*sin(x/2)^3

The graph:

Piecewise:

The solution

Derivative of tan(3x)sin(x/2)^3