Derivative of sec(2x)

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sec(2*x)

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

sec(2*x)

Detail solution

Rewrite the function to be differentiated:

$\sec{\left(2 x \right)} = \frac{1}{\cos{\left(2 x \right)}}$
Let $u = \cos{\left(2 x \right)}$ .
Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
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.
    1. 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:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

2*sec(2*x)*tan(2*x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /         2     \                  
8*\5 + 6*tan (2*x)/*sec(2*x)*tan(2*x)

$$8 \left(6 \tan^{2}{\left(2 x \right)} + 5\right) \tan{\left(2 x \right)} \sec{\left(2 x \right)}$$

Simplify