Derivative of cscx-sec(2x)

The solution

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csc(x) - sec(2*x)

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

d                    
--(csc(x) - sec(2*x))
dx

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

Transform

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

   2             /       2   \               2                   /       2     \         
cot (x)*csc(x) + \1 + cot (x)/*csc(x) - 4*tan (2*x)*sec(2*x) - 4*\1 + tan (2*x)/*sec(2*x)

$$- 4 \left(\tan^{2}{\left(2 x \right)} + 1\right) \sec{\left(2 x \right)} + \left(\cot^{2}{\left(x \right)} + 1\right) \csc{\left(x \right)} - 4 \tan^{2}{\left(2 x \right)} \sec{\left(2 x \right)} + \cot^{2}{\left(x \right)} \csc{\left(x \right)}$$

Simplify

The third derivative [src]

 /   3                  3                   /       2   \                    /       2     \                  \
-\cot (x)*csc(x) + 8*tan (2*x)*sec(2*x) + 5*\1 + cot (x)/*cot(x)*csc(x) + 40*\1 + tan (2*x)/*sec(2*x)*tan(2*x)/

$$- (40 \left(\tan^{2}{\left(2 x \right)} + 1\right) \tan{\left(2 x \right)} \sec{\left(2 x \right)} + 5 \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} \csc{\left(x \right)} + 8 \tan^{3}{\left(2 x \right)} \sec{\left(2 x \right)} + \cot^{3}{\left(x \right)} \csc{\left(x \right)})$$

Simplify

The graph

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Derivative of cscx-sec(2x)

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The solution