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Derivative of:
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Identical expressions
tg^ two (x)+ one -cos(x)
tg squared (x) plus 1 minus co sinus of e of (x)
tg to the power of two (x) plus one minus co sinus of e of (x)
tg2(x)+1-cos(x)
tg2x+1-cosx
tg²(x)+1-cos(x)
tg to the power of 2(x)+1-cos(x)
tg^2x+1-cosx
Similar expressions
tg^2(x)-1-cos(x)
tg^2(x)+1+cos(x)
tg^2(x)+1-cosx

The derivative of the function/ tg^2(x)+1-cos(x)

Derivative of tg^2(x)+1-cos(x)

The solution

You have entered [src]

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

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

d /   2                \
--\tan (x) + 1 - cos(x)/
dx

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

Transform

Detail solution

Differentiate $- \cos{\left(x \right)} + \tan^{2}{\left(x \right)} + 1$ term by term:
1. Let $u = \tan{\left(x \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(x \right)}$ :
  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:
    
    $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  The result of the chain rule is:
  
  $\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
4. The derivative of the constant $1$ is zero.
5. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  So, the result is: $\sin{\left(x \right)}$
The result is: $\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \sin{\left(x \right)}$
Now simplify:

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

The answer is:

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

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)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                                    2       
               3    /       2   \      /       2   \        
-sin(x) + 8*tan (x)*\1 + tan (x)/ + 16*\1 + tan (x)/ *tan(x)

$$8 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{3}{\left(x \right)} + 16 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \tan{\left(x \right)} - \sin{\left(x \right)}$$

Simplify

The graph

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Identical expressions

Similar expressions

Derivative of tg^2(x)+1-cos(x)

The graph:

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The solution