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three *sin(tan(five *x+pi))
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Similar expressions
3*sin(tan(5*x-pi))

The derivative of the function/ 3*sin(tan(5*x+pi))

Derivative of 3sin(tan(5x+pi))

The solution

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3*sin(tan(5*x + pi))

$$3 \sin{\left(\tan{\left(5 x + \pi \right)} \right)}$$

3*sin(tan(5*x + pi))

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \tan{\left(5 x + \pi \right)}$ .
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 $\frac{d}{d x} \tan{\left(5 x + \pi \right)}$ :
  1. Rewrite the function to be differentiated:
    
    $\tan{\left(5 x + \pi \right)} = \frac{\sin{\left(5 x \right)}}{\cos{\left(5 x \right)}}$
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sin{\left(5 x \right)}$ and $g{\left(x \right)} = \cos{\left(5 x \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Let $u = 5 x$ .
    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 $\frac{d}{d x} 5 x$ :
      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: $5$
      The result of the chain rule is:
      
      $5 \cos{\left(5 x \right)}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Let $u = 5 x$ .
    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 $\frac{d}{d x} 5 x$ :
      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: $5$
      The result of the chain rule is:
      
      $- 5 \sin{\left(5 x \right)}$
    Now plug in to the quotient rule:
    
    $\frac{5 \sin^{2}{\left(5 x \right)} + 5 \cos^{2}{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}}$
  The result of the chain rule is:
  
  $\frac{\left(5 \sin^{2}{\left(5 x \right)} + 5 \cos^{2}{\left(5 x \right)}\right) \cos{\left(\tan{\left(5 x + \pi \right)} \right)}}{\cos^{2}{\left(5 x \right)}}$
So, the result is: $\frac{3 \left(5 \sin^{2}{\left(5 x \right)} + 5 \cos^{2}{\left(5 x \right)}\right) \cos{\left(\tan{\left(5 x + \pi \right)} \right)}}{\cos^{2}{\left(5 x \right)}}$
Now simplify:

$\frac{15 \cos{\left(\tan{\left(5 x \right)} \right)}}{\cos^{2}{\left(5 x \right)}}$

The answer is:

$\frac{15 \cos{\left(\tan{\left(5 x \right)} \right)}}{\cos^{2}{\left(5 x \right)}}$

The graph

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The first derivative [src]

  /         2          \                   
3*\5 + 5*tan (5*x + pi)/*cos(tan(5*x + pi))

$$3 \left(5 \tan^{2}{\left(5 x + \pi \right)} + 5\right) \cos{\left(\tan{\left(5 x + \pi \right)} \right)}$$

Simplify

The second derivative [src]

    /       2     \ //       2     \                                         \
-75*\1 + tan (5*x)/*\\1 + tan (5*x)/*sin(tan(5*x)) - 2*cos(tan(5*x))*tan(5*x)/

$$- 75 \left(\left(\tan^{2}{\left(5 x \right)} + 1\right) \sin{\left(\tan{\left(5 x \right)} \right)} - 2 \cos{\left(\tan{\left(5 x \right)} \right)} \tan{\left(5 x \right)}\right) \left(\tan^{2}{\left(5 x \right)} + 1\right)$$

Simplify

The third derivative [src]

                     /               2                                                                                                                       \
     /       2     \ |/       2     \                       2                        /       2     \                   /       2     \                       |
-375*\1 + tan (5*x)/*\\1 + tan (5*x)/ *cos(tan(5*x)) - 4*tan (5*x)*cos(tan(5*x)) - 2*\1 + tan (5*x)/*cos(tan(5*x)) + 6*\1 + tan (5*x)/*sin(tan(5*x))*tan(5*x)/

$$- 375 \left(\tan^{2}{\left(5 x \right)} + 1\right) \left(\left(\tan^{2}{\left(5 x \right)} + 1\right)^{2} \cos{\left(\tan{\left(5 x \right)} \right)} + 6 \left(\tan^{2}{\left(5 x \right)} + 1\right) \sin{\left(\tan{\left(5 x \right)} \right)} \tan{\left(5 x \right)} - 2 \left(\tan^{2}{\left(5 x \right)} + 1\right) \cos{\left(\tan{\left(5 x \right)} \right)} - 4 \cos{\left(\tan{\left(5 x \right)} \right)} \tan^{2}{\left(5 x \right)}\right)$$

Simplify

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Derivative of 3sin(tan(5x+pi))

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Piecewise:

The solution

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Similar expressions

Derivative of 3*sin(tan(5*x+pi))

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Piecewise:

The solution

Derivative of 3sin(tan(5x+pi))