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Derivative of:
Derivative of x*sqrt(3-x)
Derivative of x^6/6
Derivative of x^3-3*x
Derivative of x^2+4x
Identical expressions
(sec(x))^ two - one
(sec(x)) squared minus 1
(sec(x)) to the power of two minus one
(sec(x))2-1
secx2-1
(sec(x))²-1
(sec(x)) to the power of 2-1
secx^2-1
Similar expressions
(sec(x))^2+1

The derivative of the function/ (sec(x))^2-1

Derivative of (sec(x))^2-1

The solution

You have entered [src]

   2       
sec (x) - 1

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

d /   2       \
--\sec (x) - 1/
dx

$$\frac{d}{d x} \left(\sec^{2}{\left(x \right)} - 1\right)$$

Transform

Detail solution

Differentiate $\sec^{2}{\left(x \right)} - 1$ term by term:
1. Let $u = \sec{\left(x \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. 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:
  
  $\frac{2 \sin{\left(x \right)} \sec{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
4. The derivative of the constant $\left(-1\right) 1$ is zero.
The result 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 (sec(x))^2-1