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Identical expressions
(ln^ two)*(tg* ten *x)
(ln squared ) multiply by (tg multiply by 10 multiply by x)
(ln to the power of two) multiply by (tg multiply by ten multiply by x)
(ln2)*(tg*10*x)
ln2*tg*10*x
(ln²)*(tg*10*x)
(ln to the power of 2)*(tg*10*x)
(ln^2)(tg10x)
(ln2)(tg10x)
ln2tg10x
ln^2tg10x

The derivative of the function/ (ln^2)*(tg*10*x)

Derivative of (ln^2)(tg10*x)

The solution

You have entered [src]

   2             
log (x)*tan(10*x)

$$\log{\left(x \right)}^{2} \tan{\left(10 x \right)}$$

log(x)^2*tan(10*x)

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = \log{\left(x \right)}^{2}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \log{\left(x \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)}$ :
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  The result of the chain rule is:
  
  $\frac{2 \log{\left(x \right)}}{x}$
$g{\left(x \right)} = \tan{\left(10 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(10 x \right)} = \frac{\sin{\left(10 x \right)}}{\cos{\left(10 x \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(10 x \right)}$ and $g{\left(x \right)} = \cos{\left(10 x \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = 10 x$ .
  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 $\frac{d}{d x} 10 x$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $10$
    The result of the chain rule is:
    
    $10 \cos{\left(10 x \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = 10 x$ .
  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 $\frac{d}{d x} 10 x$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $10$
    The result of the chain rule is:
    
    $- 10 \sin{\left(10 x \right)}$
  Now plug in to the quotient rule:
  
  $\frac{10 \sin^{2}{\left(10 x \right)} + 10 \cos^{2}{\left(10 x \right)}}{\cos^{2}{\left(10 x \right)}}$
The result is: $\frac{\left(10 \sin^{2}{\left(10 x \right)} + 10 \cos^{2}{\left(10 x \right)}\right) \log{\left(x \right)}^{2}}{\cos^{2}{\left(10 x \right)}} + \frac{2 \log{\left(x \right)} \tan{\left(10 x \right)}}{x}$
Now simplify:

$\frac{2 \left(10 x \log{\left(x \right)} + \sin{\left(20 x \right)}\right) \log{\left(x \right)}}{x \left(\cos{\left(20 x \right)} + 1\right)}$

The answer is:

$\frac{2 \left(10 x \log{\left(x \right)} + \sin{\left(20 x \right)}\right) \log{\left(x \right)}}{x \left(\cos{\left(20 x \right)} + 1\right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   2    /           2      \   2*log(x)*tan(10*x)
log (x)*\10 + 10*tan (10*x)/ + ------------------
                                       x

$$\left(10 \tan^{2}{\left(10 x \right)} + 10\right) \log{\left(x \right)}^{2} + \frac{2 \log{\left(x \right)} \tan{\left(10 x \right)}}{x}$$

Simplify

The second derivative [src]

  /                               /       2      \                                                \
  |  (-1 + log(x))*tan(10*x)   20*\1 + tan (10*x)/*log(x)          2    /       2      \          |
2*|- ----------------------- + -------------------------- + 100*log (x)*\1 + tan (10*x)/*tan(10*x)|
  |              2                         x                                                      |
  \             x                                                                                 /

$$2 \left(100 \left(\tan^{2}{\left(10 x \right)} + 1\right) \log{\left(x \right)}^{2} \tan{\left(10 x \right)} + \frac{20 \left(\tan^{2}{\left(10 x \right)} + 1\right) \log{\left(x \right)}}{x} - \frac{\left(\log{\left(x \right)} - 1\right) \tan{\left(10 x \right)}}{x^{2}}\right)$$

Simplify

The third derivative [src]

  /                               /       2      \                                                                        /       2      \                 \
  |(-3 + 2*log(x))*tan(10*x)   30*\1 + tan (10*x)/*(-1 + log(x))           2    /       2      \ /         2      \   600*\1 + tan (10*x)/*log(x)*tan(10*x)|
2*|------------------------- - --------------------------------- + 1000*log (x)*\1 + tan (10*x)/*\1 + 3*tan (10*x)/ + -------------------------------------|
  |             3                               2                                                                                       x                  |
  \            x                               x                                                                                                           /

$$2 \left(1000 \left(\tan^{2}{\left(10 x \right)} + 1\right) \left(3 \tan^{2}{\left(10 x \right)} + 1\right) \log{\left(x \right)}^{2} + \frac{600 \left(\tan^{2}{\left(10 x \right)} + 1\right) \log{\left(x \right)} \tan{\left(10 x \right)}}{x} - \frac{30 \left(\log{\left(x \right)} - 1\right) \left(\tan^{2}{\left(10 x \right)} + 1\right)}{x^{2}} + \frac{\left(2 \log{\left(x \right)} - 3\right) \tan{\left(10 x \right)}}{x^{3}}\right)$$

Simplify

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Identical expressions

Derivative of (ln^2)(tg10*x)

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Derivative of (ln^2)*(tg*10*x)

The graph:

Piecewise:

The solution

Derivative of (ln^2)(tg10*x)