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Derivative of:
Derivative of 4^(2*x)
Derivative of y=2x+3
Derivative of ln(x+5)^5
Derivative of (x^2-2*x)/(3+x^2)
Limit of the function:
sin(2*x)/cos(5*x)
Identical expressions
sin(two *x)/cos(five *x)
sinus of (2 multiply by x) divide by co sinus of e of (5 multiply by x)
sinus of (two multiply by x) divide by co sinus of e of (five multiply by x)
sin(2x)/cos(5x)
sin2x/cos5x
sin(2*x) divide by cos(5*x)

The derivative of the function/ sin(2*x)/cos(5*x)

Derivative of sin(2x)/cos(5x)

The solution

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sin(2*x)
--------
cos(5*x)

$$\frac{\sin{\left(2 x \right)}}{\cos{\left(5 x \right)}}$$

sin(2*x)/cos(5*x)

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \sin{\left(2 x \right)}$ and $g{\left(x \right)} = \cos{\left(5 x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 2 x$ .
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 $\frac{d}{d x} 2 x$ :
  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: $2$
  The result of the chain rule is:
  
  $2 \cos{\left(2 x \right)}$
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:

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

$\frac{7 \cos{\left(3 x \right)} - 3 \cos{\left(7 x \right)}}{2 \cos^{2}{\left(5 x \right)}}$

The answer is:

$\frac{7 \cos{\left(3 x \right)} - 3 \cos{\left(7 x \right)}}{2 \cos^{2}{\left(5 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

2*cos(2*x)   5*sin(2*x)*sin(5*x)
---------- + -------------------
 cos(5*x)            2          
                  cos (5*x)

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

Simplify

The second derivative [src]

                 /         2     \                                
                 |    2*sin (5*x)|            20*cos(2*x)*sin(5*x)
-4*sin(2*x) + 25*|1 + -----------|*sin(2*x) + --------------------
                 |        2      |                  cos(5*x)      
                 \     cos (5*x) /                                
------------------------------------------------------------------
                             cos(5*x)

$$\frac{25 \left(\frac{2 \sin^{2}{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}} + 1\right) \sin{\left(2 x \right)} - 4 \sin{\left(2 x \right)} + \frac{20 \sin{\left(5 x \right)} \cos{\left(2 x \right)}}{\cos{\left(5 x \right)}}}{\cos{\left(5 x \right)}}$$

Simplify

The third derivative [src]

                                                                          /         2     \                  
                                                                          |    6*sin (5*x)|                  
                                                                      125*|5 + -----------|*sin(2*x)*sin(5*x)
                  /         2     \                                       |        2      |                  
                  |    2*sin (5*x)|            60*sin(2*x)*sin(5*x)       \     cos (5*x) /                  
-8*cos(2*x) + 150*|1 + -----------|*cos(2*x) - -------------------- + ---------------------------------------
                  |        2      |                  cos(5*x)                         cos(5*x)               
                  \     cos (5*x) /                                                                          
-------------------------------------------------------------------------------------------------------------
                                                   cos(5*x)

$$\frac{150 \left(\frac{2 \sin^{2}{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}} + 1\right) \cos{\left(2 x \right)} + \frac{125 \left(\frac{6 \sin^{2}{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}} + 5\right) \sin{\left(2 x \right)} \sin{\left(5 x \right)}}{\cos{\left(5 x \right)}} - \frac{60 \sin{\left(2 x \right)} \sin{\left(5 x \right)}}{\cos{\left(5 x \right)}} - 8 \cos{\left(2 x \right)}}{\cos{\left(5 x \right)}}$$

Simplify

The graph

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Identical expressions

Derivative of sin(2x)/cos(5x)

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Derivative of sin(2*x)/cos(5*x)

The graph:

Piecewise:

The solution

Derivative of sin(2x)/cos(5x)