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Similar expressions
2/3*tgx-sin*x/3

The derivative of the function/ 2/3*tgx+sin*x/3

Derivative of 2/3tgx+sinx/3

The solution

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2*tan(x)   sin(x)
-------- + ------
   3         3

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

2*tan(x)/3 + sin(x)/3

Detail solution

Differentiate $\frac{\sin{\left(x \right)}}{3} + \frac{2 \tan{\left(x \right)}}{3}$ term by term:
1. The derivative of a constant times a function is the constant times the derivative of the function.
  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:
    
    $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  So, the result is: $\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{3 \cos^{2}{\left(x \right)}}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  So, the result is: $\frac{\cos{\left(x \right)}}{3}$
The result is: $\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{3 \cos^{2}{\left(x \right)}} + \frac{\cos{\left(x \right)}}{3}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                  2   
2   cos(x)   2*tan (x)
- + ------ + ---------
3     3          3

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

Simplify

The second derivative [src]

            /       2   \       
-sin(x) + 4*\1 + tan (x)/*tan(x)
--------------------------------
               3

$$\frac{4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \sin{\left(x \right)}}{3}$$

Simplify

The third derivative [src]

                         2                          
            /       2   \         2    /       2   \
-cos(x) + 4*\1 + tan (x)/  + 8*tan (x)*\1 + tan (x)/
----------------------------------------------------
                         3

$$\frac{4 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 8 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} - \cos{\left(x \right)}}{3}$$

Simplify

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Derivative of 2/3tgx+sinx/3

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The solution

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Derivative of 2/3*tgx+sin*x/3

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Derivative of 2/3tgx+sinx/3