Derivative of csc²x+sec²x

The solution

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   2         2   
csc (x) + sec (x)

$$\csc^{2}{\left(x \right)} + \sec^{2}{\left(x \right)}$$

d /   2         2   \
--\csc (x) + sec (x)/
dx

$$\frac{d}{d x} \left(\csc^{2}{\left(x \right)} + \sec^{2}{\left(x \right)}\right)$$

Transform

Detail solution

Differentiate $\csc^{2}{\left(x \right)} + \sec^{2}{\left(x \right)}$ term by term:
1. Let $u = \csc{\left(x \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \csc{\left(x \right)}$ :
  1. Rewrite the function to be differentiated:
    
    $\csc{\left(x \right)} = \frac{1}{\sin{\left(x \right)}}$
  2. Let $u = \sin{\left(x \right)}$ .
  3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
    1. The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    The result of the chain rule is:
    
    $- \frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
  The result of the chain rule is:
  
  $- \frac{2 \cos{\left(x \right)} \csc{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
4. Let $u = \sec{\left(x \right)}$ .
5. Apply the power rule: $u^{2}$ goes to $2 u$
6. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sec{\left(x \right)}$ :
  1. Rewrite the function to be differentiated:
    
    $\sec{\left(x \right)} = \frac{1}{\cos{\left(x \right)}}$
  2. Let $u = \cos{\left(x \right)}$ .
  3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  4. 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:
    
    $\frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  The result of the chain rule is:
  
  $\frac{2 \sin{\left(x \right)} \sec{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
The result is: $\frac{2 \sin{\left(x \right)} \sec{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{2 \cos{\left(x \right)} \csc{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2                  2          
- 2*csc (x)*cot(x) + 2*sec (x)*tan(x)

$$2 \tan{\left(x \right)} \sec^{2}{\left(x \right)} - 2 \cot{\left(x \right)} \csc^{2}{\left(x \right)}$$

Simplify

The second derivative [src]

  /   2    /       2   \      2    /       2   \        2       2           2       2   \
2*\csc (x)*\1 + cot (x)/ + sec (x)*\1 + tan (x)/ + 2*cot (x)*csc (x) + 2*sec (x)*tan (x)/

$$2 \left(2 \tan^{2}{\left(x \right)} \sec^{2}{\left(x \right)} + 2 \cot^{2}{\left(x \right)} \csc^{2}{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \sec^{2}{\left(x \right)} + \left(\cot^{2}{\left(x \right)} + 1\right) \csc^{2}{\left(x \right)}\right)$$

Simplify

The third derivative [src]

  /   2       3         3       2           2    /       2   \               2    /       2   \       \
8*\sec (x)*tan (x) - cot (x)*csc (x) - 2*csc (x)*\1 + cot (x)/*cot(x) + 2*sec (x)*\1 + tan (x)/*tan(x)/

$$8 \cdot \left(\tan^{3}{\left(x \right)} \sec^{2}{\left(x \right)} - \cot^{3}{\left(x \right)} \csc^{2}{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \sec^{2}{\left(x \right)} - 2 \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} \csc^{2}{\left(x \right)}\right)$$

Simplify

The graph

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Derivative of csc²x+sec²x

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The solution