Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of x^(7/6)
Derivative of e^x-7
Derivative of x*sin(3*x)
Derivative of (x^2+4)/x
Identical expressions
sqrt(one /cos(two *x- one))
square root of (1 divide by co sinus of e of (2 multiply by x minus 1))
square root of (one divide by co sinus of e of (two multiply by x minus one))
√(1/cos(2*x-1))
sqrt(1/cos(2x-1))
sqrt1/cos2x-1
sqrt(1 divide by cos(2*x-1))
Similar expressions
sqrt(1/cos(2*x+1))

The derivative of the function/ sqrt(1/cos(2*x-1))

Derivative of sqrt(1/cos(2*x-1))

The solution

You have entered [src]

    ______________
   /      1       
  /  ------------ 
\/   cos(2*x - 1)

$$\sqrt{\frac{1}{\cos{\left(2 x - 1 \right)}}}$$

sqrt(1/cos(2*x - 1))

Detail solution

Let $u = \frac{1}{\cos{\left(2 x - 1 \right)}}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{1}{\cos{\left(2 x - 1 \right)}}$ :
1. Let $u = \cos{\left(2 x - 1 \right)}$ .
2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(2 x - 1 \right)}$ :
  1. Let $u = 2 x - 1$ .
  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 $\frac{d}{d x} \left(2 x - 1\right)$ :
    1. Differentiate $2 x - 1$ term by term:
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $2$
      2. The derivative of the constant $-1$ is zero.
      The result is: $2$
    The result of the chain rule is:
    
    $- 2 \sin{\left(2 x - 1 \right)}$
  The result of the chain rule is:
  
  $\frac{2 \sin{\left(2 x - 1 \right)}}{\cos^{2}{\left(2 x - 1 \right)}}$
The result of the chain rule is:

$\frac{\sin{\left(2 x - 1 \right)}}{\sqrt{\frac{1}{\cos{\left(2 x - 1 \right)}}} \cos^{2}{\left(2 x - 1 \right)}}$
Now simplify:

$\frac{\sin{\left(2 x - 1 \right)}}{\sqrt{\frac{1}{\cos{\left(2 x - 1 \right)}}} \cos^{2}{\left(2 x - 1 \right)}}$

The answer is:

$\frac{\sin{\left(2 x - 1 \right)}}{\sqrt{\frac{1}{\cos{\left(2 x - 1 \right)}}} \cos^{2}{\left(2 x - 1 \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    ______________             
   /      1                    
  /  ------------ *sin(2*x - 1)
\/   cos(2*x - 1)              
-------------------------------
          cos(2*x - 1)

$$\frac{\sqrt{\frac{1}{\cos{\left(2 x - 1 \right)}}} \sin{\left(2 x - 1 \right)}}{\cos{\left(2 x - 1 \right)}}$$

Simplify

The second derivative [src]

    _______________ /         2          \
   /       1        |    3*sin (-1 + 2*x)|
  /  ------------- *|2 + ----------------|
\/   cos(-1 + 2*x)  |        2           |
                    \     cos (-1 + 2*x) /

$$\left(\frac{3 \sin^{2}{\left(2 x - 1 \right)}}{\cos^{2}{\left(2 x - 1 \right)}} + 2\right) \sqrt{\frac{1}{\cos{\left(2 x - 1 \right)}}}$$

Simplify

The third derivative [src]

    _______________ /           2          \              
   /       1        |     15*sin (-1 + 2*x)|              
  /  ------------- *|14 + -----------------|*sin(-1 + 2*x)
\/   cos(-1 + 2*x)  |          2           |              
                    \       cos (-1 + 2*x) /              
----------------------------------------------------------
                      cos(-1 + 2*x)

$$\frac{\left(\frac{15 \sin^{2}{\left(2 x - 1 \right)}}{\cos^{2}{\left(2 x - 1 \right)}} + 14\right) \sqrt{\frac{1}{\cos{\left(2 x - 1 \right)}}} \sin{\left(2 x - 1 \right)}}{\cos{\left(2 x - 1 \right)}}$$

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of sqrt(1/cos(2*x-1))

The graph:

Piecewise:

The solution