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Identical expressions
two *tan(three x)+3*tan(2x)
2 multiply by tangent of (3x) plus 3 multiply by tangent of (2x)
two multiply by tangent of (three x) plus 3 multiply by tangent of (2x)
2tan(3x)+3tan(2x)
2tan3x+3tan2x
Similar expressions
2*tan(3x)-3*tan(2x)

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

Derivative of 2tan(3x)+3tan(2x)

The solution

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2*tan(3*x) + 3*tan(2*x)

$$3 \tan{\left(2 x \right)} + 2 \tan{\left(3 x \right)}$$

2*tan(3*x) + 3*tan(2*x)

Detail solution

Differentiate $3 \tan{\left(2 x \right)} + 2 \tan{\left(3 x \right)}$ term by term:
1. The derivative of a constant times a function is the constant times the derivative of the function.
  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.
        
        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.
        
        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:
    
    $\frac{3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}}$
  So, the result is: $\frac{2 \left(3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}\right)}{\cos^{2}{\left(3 x \right)}}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = 2 x$ .
  2. $\frac{d}{d u} \tan{\left(u \right)} = \frac{1}{\cos^{2}{\left(u \right)}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
    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: $2$
    The result of the chain rule is:
    
    $\frac{2}{\cos^{2}{\left(2 x \right)}}$
  So, the result is: $\frac{3 \left(2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}\right)}{\cos^{2}{\left(2 x \right)}}$
The result is: $\frac{3 \left(2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}\right)}{\cos^{2}{\left(2 x \right)}} + \frac{2 \left(3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}\right)}{\cos^{2}{\left(3 x \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

          2             2     
12 + 6*tan (2*x) + 6*tan (3*x)

$$6 \tan^{2}{\left(2 x \right)} + 6 \tan^{2}{\left(3 x \right)} + 12$$

Simplify

The second derivative [src]

   /  /       2     \              /       2     \         \
12*\2*\1 + tan (2*x)/*tan(2*x) + 3*\1 + tan (3*x)/*tan(3*x)/

$$12 \left(2 \left(\tan^{2}{\left(2 x \right)} + 1\right) \tan{\left(2 x \right)} + 3 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)}\right)$$

Simplify

3-я производная [src]

   /                 2                    2                                                             \
   |  /       2     \      /       2     \         2      /       2     \         2      /       2     \|
12*\4*\1 + tan (2*x)/  + 9*\1 + tan (3*x)/  + 8*tan (2*x)*\1 + tan (2*x)/ + 18*tan (3*x)*\1 + tan (3*x)//

$$12 \left(4 \left(\tan^{2}{\left(2 x \right)} + 1\right)^{2} + 8 \left(\tan^{2}{\left(2 x \right)} + 1\right) \tan^{2}{\left(2 x \right)} + 9 \left(\tan^{2}{\left(3 x \right)} + 1\right)^{2} + 18 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan^{2}{\left(3 x \right)}\right)$$

Simplify

The third derivative [src]

   /                 2                    2                                                             \
   |  /       2     \      /       2     \         2      /       2     \         2      /       2     \|
12*\4*\1 + tan (2*x)/  + 9*\1 + tan (3*x)/  + 8*tan (2*x)*\1 + tan (2*x)/ + 18*tan (3*x)*\1 + tan (3*x)//

Simplify

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Derivative of 2tan(3x)+3tan(2x)

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Piecewise:

The solution

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Derivative of 2*tan(3x)+3*tan(2x)

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The solution

Derivative of 2tan(3x)+3tan(2x)