Derivative of 3sin(tg(5x+n))

The solution

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

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

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

Detail solution

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

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

The answer is:

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

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of 3sin(tg(5x+n))

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