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Identical expressions
tgx+lnx+(x^ four / four)
tgx plus lnx plus (x to the power of 4 divide by 4)
tgx plus lnx plus (x to the power of four divide by four)
tgx+lnx+(x4/4)
tgx+lnx+x4/4
tgx+lnx+(x⁴/4)
tgx+lnx+x^4/4
tgx+lnx+(x^4 divide by 4)
Similar expressions
tgx-lnx+(x^4/4)
tgx+lnx-(x^4/4)

The derivative of the function/ tgx+lnx+(x^4/4)

Derivative of tgx+lnx+(x^4/4)

The solution

You have entered [src]

                   4
                  x 
tan(x) + log(x) + --
                  4

$$\frac{x^{4}}{4} + \left(\log{\left(x \right)} + \tan{\left(x \right)}\right)$$

tan(x) + log(x) + x^4/4

Detail solution

Differentiate $\frac{x^{4}}{4} + \left(\log{\left(x \right)} + \tan{\left(x \right)}\right)$ term by term:
1. Differentiate $\log{\left(x \right)} + \tan{\left(x \right)}$ term by term:
  1. Rewrite the function to be differentiated:
    
    $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = \cos{\left(x \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    Now plug in to the quotient rule:
    
    $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  3. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  The result is: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{1}{x}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $x^{4}$ goes to $4 x^{3}$
  So, the result is: $x^{3}$
The result is: $x^{3} + \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{1}{x}$
Now simplify:

$\frac{x^{4} + \frac{x}{\cos^{2}{\left(x \right)}} + 1}{x}$

The answer is:

$\frac{x^{4} + \frac{x}{\cos^{2}{\left(x \right)}} + 1}{x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    1    3      2   
1 + - + x  + tan (x)
    x

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                  2                                \
  |1    /       2   \               2    /       2   \|
2*|-- + \1 + tan (x)/  + 3*x + 2*tan (x)*\1 + tan (x)/|
  | 3                                                 |
  \x                                                  /

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

Simplify

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Derivative of tgx+lnx+(x^4/4)

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The solution