You entered:

tan(x)-1/cos(x)

What you mean?

(tan(x) - 1*1)/cos(x)

(tan(x) - 1*1)*x/cos(x)

tan(x - 1*1)/cos(x)

tan(x - 1*1)*x/cos(x)

tan(x) - 1/cos(x)

tan(x) - x/cos(x)

tan(x) - x/cos(x)

Derivative of tan(x)-1/cos(x)

The solution

You have entered [src]

             1   
tan(x) - 1*------
           cos(x)

$$\tan{\left(x \right)} - 1 \cdot \frac{1}{\cos{\left(x \right)}}$$

d /             1   \
--|tan(x) - 1*------|
dx\           cos(x)/

$$\frac{d}{d x} \left(\tan{\left(x \right)} - 1 \cdot \frac{1}{\cos{\left(x \right)}}\right)$$

Transform

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

               2        3                                        
  /       2   \    6*sin (x)   5*sin(x)        2    /       2   \
2*\1 + tan (x)/  - --------- - -------- + 4*tan (x)*\1 + tan (x)/
                       4          2                              
                    cos (x)    cos (x)

$$4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} - \frac{6 \sin^{3}{\left(x \right)}}{\cos^{4}{\left(x \right)}} - \frac{5 \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$$

Simplify

The graph

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Derivative of tan(x)-1/cos(x)

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