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Identical expressions
y=tg^2x+ln(sinx)
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y=tg^2x+lnsinx
Similar expressions
y=tg^2x-ln(sinx)

The derivative of the function/ y=tg^2x+ln(sinx)

Derivative of y=tg^2x+ln(sinx)

The solution

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   2                 
tan (x) + log(sin(x))

$$\tan^{2}{\left(x \right)} + \log{\left(\sin{\left(x \right)} \right)}$$

d /   2                 \
--\tan (x) + log(sin(x))/
dx

$$\frac{d}{d x} \left(\tan^{2}{\left(x \right)} + \log{\left(\sin{\left(x \right)} \right)}\right)$$

Transform

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

cos(x)   /         2   \       
------ + \2 + 2*tan (x)/*tan(x)
sin(x)

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

Simplify

The second derivative [src]

                    2      2                             
       /       2   \    cos (x)        2    /       2   \
-1 + 2*\1 + tan (x)/  - ------- + 4*tan (x)*\1 + tan (x)/
                           2                             
                        sin (x)

$$4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} - 1 - \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}$$

Simplify

The third derivative [src]

  /   3                                                        2       \
  |cos (x)   cos(x)        3    /       2   \     /       2   \        |
2*|------- + ------ + 4*tan (x)*\1 + tan (x)/ + 8*\1 + tan (x)/ *tan(x)|
  |   3      sin(x)                                                    |
  \sin (x)                                                             /

$$2 \cdot \left(4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{3}{\left(x \right)} + 8 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \tan{\left(x \right)} + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{\cos^{3}{\left(x \right)}}{\sin^{3}{\left(x \right)}}\right)$$

Simplify

3-th derivative [src]

  /   3                                                        2       \
  |cos (x)   cos(x)        3    /       2   \     /       2   \        |
2*|------- + ------ + 4*tan (x)*\1 + tan (x)/ + 8*\1 + tan (x)/ *tan(x)|
  |   3      sin(x)                                                    |
  \sin (x)                                                             /

Simplify

The graph

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Derivative of y=tg^2x+ln(sinx)

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The solution