Derivative of 2cos(x)-6sec(x)+3

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2*cos(x) - 6*sec(x) + 3

$$2 \cos{\left(x \right)} - 6 \sec{\left(x \right)} + 3$$

d                          
--(2*cos(x) - 6*sec(x) + 3)
dx

$$\frac{d}{d x} \left(2 \cos{\left(x \right)} - 6 \sec{\left(x \right)} + 3\right)$$

Transform

Detail solution

Differentiate $2 \cos{\left(x \right)} - 6 \sec{\left(x \right)} + 3$ term by term:
1. 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: $- 2 \sin{\left(x \right)}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Rewrite the function to be differentiated:
      
      $\sec{\left(x \right)} = \frac{1}{\cos{\left(x \right)}}$
    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:
      
      $\frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
    So, the result is: $\frac{6 \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  So, the result is: $- \frac{6 \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
3. The derivative of the constant $3$ is zero.
The result is: $- 2 \sin{\left(x \right)} - \frac{6 \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

-2*sin(x) - 6*sec(x)*tan(x)

$$- 6 \tan{\left(x \right)} \sec{\left(x \right)} - 2 \sin{\left(x \right)}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /       3                /       2   \                       \
2*\- 3*tan (x)*sec(x) - 15*\1 + tan (x)/*sec(x)*tan(x) + sin(x)/

$$2 \left(- 3 \tan^{3}{\left(x \right)} \sec{\left(x \right)} - 15 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \sec{\left(x \right)} + \sin{\left(x \right)}\right)$$

Simplify

The graph

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Derivative of 2cos(x)-6sec(x)+3

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The solution