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Derivative of (5*x-4)*(2*x^4-7*x+1)
Identical expressions
two ^(tan4*x)*cos(lgx)
2 to the power of ( tangent of 4 multiply by x) multiply by co sinus of e of (lgx)
two to the power of ( tangent of 4 multiply by x) multiply by co sinus of e of (lgx)
2(tan4*x)*cos(lgx)
2tan4*x*coslgx
2^(tan4x)cos(lgx)
2(tan4x)cos(lgx)
2tan4xcoslgx
2^tan4xcoslgx

The derivative of the function/ 2^(tan4*x)*cos(lgx)

Derivative of 2^(tan4x)cos(lgx)

The solution

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 tan(4*x)            
2        *cos(log(x))

$$2^{\tan{\left(4 x \right)}} \cos{\left(\log{\left(x \right)} \right)}$$

2^tan(4*x)*cos(log(x))

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = 2^{\tan{\left(4 x \right)}}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \tan{\left(4 x \right)}$ .
2. $\frac{d}{d u} 2^{u} = 2^{u} \log{\left(2 \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(4 x \right)}$ :
  1. Rewrite the function to be differentiated:
    
    $\tan{\left(4 x \right)} = \frac{\sin{\left(4 x \right)}}{\cos{\left(4 x \right)}}$
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sin{\left(4 x \right)}$ and $g{\left(x \right)} = \cos{\left(4 x \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Let $u = 4 x$ .
    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 $\frac{d}{d x} 4 x$ :
      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: $4$
      The result of the chain rule is:
      
      $4 \cos{\left(4 x \right)}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Let $u = 4 x$ .
    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 $\frac{d}{d x} 4 x$ :
      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: $4$
      The result of the chain rule is:
      
      $- 4 \sin{\left(4 x \right)}$
    Now plug in to the quotient rule:
    
    $\frac{4 \sin^{2}{\left(4 x \right)} + 4 \cos^{2}{\left(4 x \right)}}{\cos^{2}{\left(4 x \right)}}$
  The result of the chain rule is:
  
  $\frac{2^{\tan{\left(4 x \right)}} \left(4 \sin^{2}{\left(4 x \right)} + 4 \cos^{2}{\left(4 x \right)}\right) \log{\left(2 \right)}}{\cos^{2}{\left(4 x \right)}}$
$g{\left(x \right)} = \cos{\left(\log{\left(x \right)} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \log{\left(x \right)}$ .
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 $\frac{d}{d x} \log{\left(x \right)}$ :
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  The result of the chain rule is:
  
  $- \frac{\sin{\left(\log{\left(x \right)} \right)}}{x}$
The result is: $\frac{2^{\tan{\left(4 x \right)}} \left(4 \sin^{2}{\left(4 x \right)} + 4 \cos^{2}{\left(4 x \right)}\right) \log{\left(2 \right)} \cos{\left(\log{\left(x \right)} \right)}}{\cos^{2}{\left(4 x \right)}} - \frac{2^{\tan{\left(4 x \right)}} \sin{\left(\log{\left(x \right)} \right)}}{x}$
Now simplify:

$\frac{4 \cdot 2^{\tan{\left(4 x \right)}} \log{\left(2 \right)} \cos{\left(\log{\left(x \right)} \right)}}{\cos^{2}{\left(4 x \right)}} - \frac{2^{\tan{\left(4 x \right)}} \sin{\left(\log{\left(x \right)} \right)}}{x}$

The answer is:

$\frac{4 \cdot 2^{\tan{\left(4 x \right)}} \log{\left(2 \right)} \cos{\left(\log{\left(x \right)} \right)}}{\cos^{2}{\left(4 x \right)}} - \frac{2^{\tan{\left(4 x \right)}} \sin{\left(\log{\left(x \right)} \right)}}{x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   tan(4*x)                                                             
  2        *sin(log(x))    tan(4*x) /         2     \                   
- --------------------- + 2        *\4 + 4*tan (4*x)/*cos(log(x))*log(2)
            x

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

Simplify

The second derivative [src]

          /                               /       2     \                                                                                                 \
 tan(4*x) |-cos(log(x)) + sin(log(x))   8*\1 + tan (4*x)/*log(2)*sin(log(x))      /       2     \ /             /       2     \       \                   |
2        *|-------------------------- - ------------------------------------ + 16*\1 + tan (4*x)/*\2*tan(4*x) + \1 + tan (4*x)/*log(2)/*cos(log(x))*log(2)|
          |             2                                x                                                                                                |
          \            x                                                                                                                                  /

$$2^{\tan{\left(4 x \right)}} \left(16 \left(\left(\tan^{2}{\left(4 x \right)} + 1\right) \log{\left(2 \right)} + 2 \tan{\left(4 x \right)}\right) \left(\tan^{2}{\left(4 x \right)} + 1\right) \log{\left(2 \right)} \cos{\left(\log{\left(x \right)} \right)} - \frac{8 \left(\tan^{2}{\left(4 x \right)} + 1\right) \log{\left(2 \right)} \sin{\left(\log{\left(x \right)} \right)}}{x} + \frac{\sin{\left(\log{\left(x \right)} \right)} - \cos{\left(\log{\left(x \right)} \right)}}{x^{2}}\right)$$

Simplify

The third derivative [src]

          /                                    /       2     \                                                          /                                 2                                            \                         /       2     \ /             /       2     \       \                   \
 tan(4*x) |  -3*cos(log(x)) + sin(log(x))   12*\1 + tan (4*x)/*(-cos(log(x)) + sin(log(x)))*log(2)      /       2     \ |         2        /       2     \     2        /       2     \                |                      48*\1 + tan (4*x)/*\2*tan(4*x) + \1 + tan (4*x)/*log(2)/*log(2)*sin(log(x))|
2        *|- ---------------------------- + ------------------------------------------------------ + 64*\1 + tan (4*x)/*\2 + 6*tan (4*x) + \1 + tan (4*x)/ *log (2) + 6*\1 + tan (4*x)/*log(2)*tan(4*x)/*cos(log(x))*log(2) - ---------------------------------------------------------------------------|
          |                3                                           2                                                                                                                                                                                           x                                     |
          \               x                                           x                                                                                                                                                                                                                                  /

$$2^{\tan{\left(4 x \right)}} \left(64 \left(\tan^{2}{\left(4 x \right)} + 1\right) \left(\left(\tan^{2}{\left(4 x \right)} + 1\right)^{2} \log{\left(2 \right)}^{2} + 6 \left(\tan^{2}{\left(4 x \right)} + 1\right) \log{\left(2 \right)} \tan{\left(4 x \right)} + 6 \tan^{2}{\left(4 x \right)} + 2\right) \log{\left(2 \right)} \cos{\left(\log{\left(x \right)} \right)} - \frac{48 \left(\left(\tan^{2}{\left(4 x \right)} + 1\right) \log{\left(2 \right)} + 2 \tan{\left(4 x \right)}\right) \left(\tan^{2}{\left(4 x \right)} + 1\right) \log{\left(2 \right)} \sin{\left(\log{\left(x \right)} \right)}}{x} + \frac{12 \left(\sin{\left(\log{\left(x \right)} \right)} - \cos{\left(\log{\left(x \right)} \right)}\right) \left(\tan^{2}{\left(4 x \right)} + 1\right) \log{\left(2 \right)}}{x^{2}} - \frac{\sin{\left(\log{\left(x \right)} \right)} - 3 \cos{\left(\log{\left(x \right)} \right)}}{x^{3}}\right)$$

Simplify

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Derivative of 2^(tan4x)cos(lgx)

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Piecewise:

The solution

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Identical expressions

Derivative of 2^(tan4*x)*cos(lgx)

The graph:

Piecewise:

The solution

Derivative of 2^(tan4x)cos(lgx)