Derivative of sec(x)

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sec(x)

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

sec(x)

Detail solution

Rewrite the function to be differentiated:

$\sec{\left(x \right)} = \frac{1}{\cos{\left(x \right)}}$
Let $u = \cos{\left(x \right)}$ .
Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
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 answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

sec(x)*tan(x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph