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1/2*tan(x)^(2)+log(cos(x))

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The derivative of the function/ 1/2*tan(x)^(2)+log(cos(x))

Derivative of 1/2*tan(x)^(2)+log(cos(x))

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   2                 
tan (x)              
------- + log(cos(x))
   2
$$\log{\left(\cos{\left(x \right)} \right)} + \frac{\tan^{2}{\left(x \right)}}{2}$$ 
tan(x)^2/2 + log(cos(x))
Detail solution

  1. Differentiate term by term:

    1. The derivative of a constant times a function is the constant times the derivative of the function.

      1. Let .

      2. Apply the power rule: goes to

      3. Then, apply the chain rule. Multiply by :

        1. Rewrite the function to be differentiated:

        2. Apply the quotient rule, which is:

          and .

          To find :

          1. The derivative of sine is cosine:

          To find :

          1. The derivative of cosine is negative sine:

          Now plug in to the quotient rule:

        The result of the chain rule is:

      So, the result is:

    2. Let .

    3. The derivative of is .

    4. Then, apply the chain rule. Multiply by :

      1. The derivative of cosine is negative sine:

      The result of the chain rule is:

    The result is:

  2. Now simplify:

The answer is:

The graph
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
/         2   \                
\2 + 2*tan (x)/*tan(x)   sin(x)
---------------------- - ------
          2              cos(x)
$$\frac{\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)}}{2} - \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$$
Simplify
The second derivative [src] 
                  2      2                             
     /       2   \    sin (x)        2    /       2   \
-1 + \1 + tan (x)/  - ------- + 2*tan (x)*\1 + tan (x)/
                         2                             
                      cos (x)
$$\left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} - \frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - 1$$
Simplify
The third derivative [src] 
  /              3                                               2       \
  |  sin(x)   sin (x)        3    /       2   \     /       2   \        |
2*|- ------ - ------- + 2*tan (x)*\1 + tan (x)/ + 4*\1 + tan (x)/ *tan(x)|
  |  cos(x)      3                                                       |
  \           cos (x)                                                    /
$$2 \left(4 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \tan{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{3}{\left(x \right)} - \frac{\sin^{3}{\left(x \right)}}{\cos^{3}{\left(x \right)}} - \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}\right)$$
Simplify
The graph
Derivative of 1/2*tan(x)^(2)+log(cos(x))
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