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Derivative of:
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Identical expressions
y=(tg(5x- two))^(one / two)
y equally (tg(5x minus 2)) to the power of (1 divide by 2)
y equally (tg(5x minus two)) to the power of (one divide by two)
y=(tg(5x-2))(1/2)
y=tg5x-21/2
y=tg5x-2^1/2
y=(tg(5x-2))^(1 divide by 2)
Similar expressions
y=(tg(5x+2))^(1/2)

The derivative of the function/ y=(tg(5x-2))^(1/2)

Derivative of y=(tg(5x-2))^(1/2)

The solution

You have entered [src]

  ______________
\/ tan(5*x - 2)

$$\sqrt{\tan{\left(5 x - 2 \right)}}$$

d /  ______________\
--\\/ tan(5*x - 2) /
dx

$$\frac{d}{d x} \sqrt{\tan{\left(5 x - 2 \right)}}$$

Transform

Detail solution

Let $u = \tan{\left(5 x - 2 \right)}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(5 x - 2 \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(5 x - 2 \right)} = \frac{\sin{\left(5 x - 2 \right)}}{\cos{\left(5 x - 2 \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(5 x - 2 \right)}$ and $g{\left(x \right)} = \cos{\left(5 x - 2 \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = 5 x - 2$ .
  2. The derivative of sine is cosine:
    
    $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(5 x - 2\right)$ :
    1. Differentiate $5 x - 2$ 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: $5$
      2. The derivative of the constant $\left(-1\right) 2$ is zero.
      The result is: $5$
    The result of the chain rule is:
    
    $5 \cos{\left(5 x - 2 \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = 5 x - 2$ .
  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 $\frac{d}{d x} \left(5 x - 2\right)$ :
    1. Differentiate $5 x - 2$ 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: $5$
      2. The derivative of the constant $\left(-1\right) 2$ is zero.
      The result is: $5$
    The result of the chain rule is:
    
    $- 5 \sin{\left(5 x - 2 \right)}$
  Now plug in to the quotient rule:
  
  $\frac{5 \sin^{2}{\left(5 x - 2 \right)} + 5 \cos^{2}{\left(5 x - 2 \right)}}{\cos^{2}{\left(5 x - 2 \right)}}$
The result of the chain rule is:

$\frac{5 \sin^{2}{\left(5 x - 2 \right)} + 5 \cos^{2}{\left(5 x - 2 \right)}}{2 \cos^{2}{\left(5 x - 2 \right)} \sqrt{\tan{\left(5 x - 2 \right)}}}$
Now simplify:

$\frac{5}{2 \cos^{2}{\left(5 x - 2 \right)} \sqrt{\tan{\left(5 x - 2 \right)}}}$

The answer is:

$\frac{5}{2 \cos^{2}{\left(5 x - 2 \right)} \sqrt{\tan{\left(5 x - 2 \right)}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

         2         
5   5*tan (5*x - 2)
- + ---------------
2          2       
-------------------
    ______________ 
  \/ tan(5*x - 2)

$$\frac{\frac{5 \tan^{2}{\left(5 x - 2 \right)}}{2} + \frac{5}{2}}{\sqrt{\tan{\left(5 x - 2 \right)}}}$$

Simplify

The second derivative [src]

   /       2          \ /                             2          \
   |1   tan (-2 + 5*x)| |    _______________   1 + tan (-2 + 5*x)|
25*|- + --------------|*|4*\/ tan(-2 + 5*x)  - ------------------|
   \4         4       / |                          3/2           |
                        \                       tan   (-2 + 5*x) /

$$25 \left(- \frac{\tan^{2}{\left(5 x - 2 \right)} + 1}{\tan^{\frac{3}{2}}{\left(5 x - 2 \right)}} + 4 \sqrt{\tan{\left(5 x - 2 \right)}}\right) \left(\frac{\tan^{2}{\left(5 x - 2 \right)}}{4} + \frac{1}{4}\right)$$

Simplify

The third derivative [src]

                         /                                                                     2\
    /       2          \ |                        /       2          \     /       2          \ |
    |1   tan (-2 + 5*x)| |      3/2             4*\1 + tan (-2 + 5*x)/   3*\1 + tan (-2 + 5*x)/ |
125*|- + --------------|*|16*tan   (-2 + 5*x) - ---------------------- + -----------------------|
    \8         8       / |                          _______________             5/2             |
                         \                        \/ tan(-2 + 5*x)           tan   (-2 + 5*x)   /

$$125 \left(\frac{\tan^{2}{\left(5 x - 2 \right)}}{8} + \frac{1}{8}\right) \left(\frac{3 \left(\tan^{2}{\left(5 x - 2 \right)} + 1\right)^{2}}{\tan^{\frac{5}{2}}{\left(5 x - 2 \right)}} - \frac{4 \left(\tan^{2}{\left(5 x - 2 \right)} + 1\right)}{\sqrt{\tan{\left(5 x - 2 \right)}}} + 16 \tan^{\frac{3}{2}}{\left(5 x - 2 \right)}\right)$$

Simplify

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Derivative of y=(tg(5x-2))^(1/2)

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The solution