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Limit of the function:
sin(10*x)/tan(5*x)
Identical expressions
sin(ten *x)/tan(five *x)
sinus of (10 multiply by x) divide by tangent of (5 multiply by x)
sinus of (ten multiply by x) divide by tangent of (five multiply by x)
sin(10x)/tan(5x)
sin10x/tan5x
sin(10*x) divide by tan(5*x)

The derivative of the function/ sin(10*x)/tan(5*x)

Derivative of sin(10x)/tan(5x)

The solution

You have entered [src]

sin(10*x)
---------
 tan(5*x)

$$\frac{\sin{\left(10 x \right)}}{\tan{\left(5 x \right)}}$$

sin(10*x)/tan(5*x)

Detail solution

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 $g{\left(x \right)} = \tan{\left(5 x \right)}$ .

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. Rewrite the function to be differentiated:
  
  $\tan{\left(5 x \right)} = \frac{\sin{\left(5 x \right)}}{\cos{\left(5 x \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(5 x \right)}$ and $g{\left(x \right)} = \cos{\left(5 x \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = 5 x$ .
  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 $\frac{d}{d x} 5 x$ :
    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: $5$
    The result of the chain rule is:
    
    $5 \cos{\left(5 x \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = 5 x$ .
  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 $\frac{d}{d x} 5 x$ :
    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: $5$
    The result of the chain rule is:
    
    $- 5 \sin{\left(5 x \right)}$
  Now plug in to the quotient rule:
  
  $\frac{5 \sin^{2}{\left(5 x \right)} + 5 \cos^{2}{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}}$
Now plug in to the quotient rule:

$\frac{- \frac{\left(5 \sin^{2}{\left(5 x \right)} + 5 \cos^{2}{\left(5 x \right)}\right) \sin{\left(10 x \right)}}{\cos^{2}{\left(5 x \right)}} + 10 \cos{\left(10 x \right)} \tan{\left(5 x \right)}}{\tan^{2}{\left(5 x \right)}}$
Now simplify:

$- 10 \sin{\left(10 x \right)}$

The answer is:

$- 10 \sin{\left(10 x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

               /          2     \          
10*cos(10*x)   \-5 - 5*tan (5*x)/*sin(10*x)
------------ + ----------------------------
  tan(5*x)                 2               
                        tan (5*x)

$$\frac{\left(- 5 \tan^{2}{\left(5 x \right)} - 5\right) \sin{\left(10 x \right)}}{\tan^{2}{\left(5 x \right)}} + \frac{10 \cos{\left(10 x \right)}}{\tan{\left(5 x \right)}}$$

Simplify

The second derivative [src]

   /                               /            2     \               /       2     \          \
   |               /       2     \ |     1 + tan (5*x)|             2*\1 + tan (5*x)/*cos(10*x)|
50*|-2*sin(10*x) + \1 + tan (5*x)/*|-1 + -------------|*sin(10*x) - ---------------------------|
   |                               |          2       |                       tan(5*x)         |
   \                               \       tan (5*x)  /                                        /
------------------------------------------------------------------------------------------------
                                            tan(5*x)

$$\frac{50 \left(\left(\frac{\tan^{2}{\left(5 x \right)} + 1}{\tan^{2}{\left(5 x \right)}} - 1\right) \left(\tan^{2}{\left(5 x \right)} + 1\right) \sin{\left(10 x \right)} - \frac{2 \left(\tan^{2}{\left(5 x \right)} + 1\right) \cos{\left(10 x \right)}}{\tan{\left(5 x \right)}} - 2 \sin{\left(10 x \right)}\right)}{\tan{\left(5 x \right)}}$$

Simplify

The third derivative [src]

    /                                                                                                                                        /            2     \          \
    |                                                                                                                        /       2     \ |     1 + tan (5*x)|          |
    |  /                                   2                    3\                                                         6*\1 + tan (5*x)/*|-1 + -------------|*cos(10*x)|
    |  |                    /       2     \      /       2     \ |                             /       2     \                               |          2       |          |
    |  |         2        5*\1 + tan (5*x)/    3*\1 + tan (5*x)/ |             4*cos(10*x)   6*\1 + tan (5*x)/*sin(10*x)                     \       tan (5*x)  /          |
250*|- |2 + 2*tan (5*x) - ------------------ + ------------------|*sin(10*x) - ----------- + --------------------------- + ------------------------------------------------|
    |  |                         2                    4          |               tan(5*x)                2                                     tan(5*x)                    |
    \  \                      tan (5*x)            tan (5*x)     /                                    tan (5*x)                                                            /

$$250 \left(\frac{6 \left(\frac{\tan^{2}{\left(5 x \right)} + 1}{\tan^{2}{\left(5 x \right)}} - 1\right) \left(\tan^{2}{\left(5 x \right)} + 1\right) \cos{\left(10 x \right)}}{\tan{\left(5 x \right)}} + \frac{6 \left(\tan^{2}{\left(5 x \right)} + 1\right) \sin{\left(10 x \right)}}{\tan^{2}{\left(5 x \right)}} - \left(\frac{3 \left(\tan^{2}{\left(5 x \right)} + 1\right)^{3}}{\tan^{4}{\left(5 x \right)}} - \frac{5 \left(\tan^{2}{\left(5 x \right)} + 1\right)^{2}}{\tan^{2}{\left(5 x \right)}} + 2 \tan^{2}{\left(5 x \right)} + 2\right) \sin{\left(10 x \right)} - \frac{4 \cos{\left(10 x \right)}}{\tan{\left(5 x \right)}}\right)$$

Simplify

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Identical expressions

Derivative of sin(10x)/tan(5x)

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Derivative of sin(10*x)/tan(5*x)

The graph:

Piecewise:

The solution

Derivative of sin(10x)/tan(5x)