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The derivative of the function/ sec²x

Derivative of sec²x

The solution

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   2   
sec (x)

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

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

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

Transform

Detail solution

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

$\frac{2 \sin{\left(x \right)} \sec{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
Now simplify:

$\frac{2 \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}}$

The answer is:

$\frac{2 \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

     2    /         2   \       
8*sec (x)*\2 + 3*tan (x)/*tan(x)

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

Simplify

The graph

Derivative of sec²x