Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of 3*x^2-1/x^3
Derivative of (1+cosx)/(1-cosx)
Derivative of (1/2)*x
Derivative of 1-1/x
Identical expressions
cos(x+(two *pi/ three))-tan(x-(pi/ four))
co sinus of e of (x plus (2 multiply by Pi divide by 3)) minus tangent of (x minus ( Pi divide by 4))
co sinus of e of (x plus (two multiply by Pi divide by three)) minus tangent of (x minus ( Pi divide by four))
cos(x+(2pi/3))-tan(x-(pi/4))
cosx+2pi/3-tanx-pi/4
cos(x+(2*pi divide by 3))-tan(x-(pi divide by 4))
Similar expressions
cos(x+(2*pi/3))+tan(x-(pi/4))
cos(x+(2*pi/3))-tan(x+(pi/4))
cos(x-(2*pi/3))-tan(x-(pi/4))

The derivative of the function/ cos(x+(2*pi/3))-tan(x-(pi/4))

Derivative of cos(x+(2*pi/3))-tan(x-(pi/4))

The solution

You have entered [src]

   /    2*pi\      /    pi\
cos|x + ----| - tan|x - --|
   \     3  /      \    4 /

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

cos(x + (2*pi)/3) - tan(x - pi/4)

Detail solution

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

$\frac{2 \left(- \frac{\sin{\left(3 x + \frac{\pi}{6} \right)}}{4} - \frac{\cos{\left(x + \frac{\pi}{6} \right)}}{2} + \frac{\cos{\left(x + \frac{\pi}{3} \right)}}{4} - 1\right)}{\sin{\left(2 x \right)} + 1}$

The answer is:

$\frac{2 \left(- \frac{\sin{\left(3 x + \frac{\pi}{6} \right)}}{4} - \frac{\cos{\left(x + \frac{\pi}{6} \right)}}{2} + \frac{\cos{\left(x + \frac{\pi}{3} \right)}}{4} - 1\right)}{\sin{\left(2 x \right)} + 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        2/    pi\      /    2*pi\
-1 - tan |x - --| - sin|x + ----|
         \    4 /      \     3  /

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

Simplify

The second derivative [src]

  /       2/    pi\\    /    pi\      /    pi\
2*|1 + cot |x + --||*cot|x + --| + sin|x + --|
  \        \    4 //    \    4 /      \    6 /

$$2 \left(\cot^{2}{\left(x + \frac{\pi}{4} \right)} + 1\right) \cot{\left(x + \frac{\pi}{4} \right)} + \sin{\left(x + \frac{\pi}{6} \right)}$$

Simplify

3-я производная [src]

                      2                                                  
    /       2/    pi\\         2/    pi\ /       2/    pi\\      /    pi\
- 2*|1 + cot |x + --||  - 4*cot |x + --|*|1 + cot |x + --|| + cos|x + --|
    \        \    4 //          \    4 / \        \    4 //      \    6 /

$$- 2 \left(\cot^{2}{\left(x + \frac{\pi}{4} \right)} + 1\right)^{2} - 4 \left(\cot^{2}{\left(x + \frac{\pi}{4} \right)} + 1\right) \cot^{2}{\left(x + \frac{\pi}{4} \right)} + \cos{\left(x + \frac{\pi}{6} \right)}$$

Simplify

The third derivative [src]

                      2                                                  
    /       2/    pi\\         2/    pi\ /       2/    pi\\      /    pi\
- 2*|1 + cot |x + --||  - 4*cot |x + --|*|1 + cot |x + --|| + cos|x + --|
    \        \    4 //          \    4 / \        \    4 //      \    6 /

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of cos(x+(2*pi/3))-tan(x-(pi/4))

The graph:

Piecewise:

The solution