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The derivative of the function/ (3x-2sinx)/(5cos^2x)

Derivative of (3x-2sinx)/(5cos^2x)

The solution

You have entered [src]

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

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

(3*x - 2*sin(x))/((5*cos(x)^2))

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 3 x - 2 \sin{\left(x \right)}$ and $g{\left(x \right)} = 5 \cos^{2}{\left(x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $3 x - 2 \sin{\left(x \right)}$ term by term:
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    So, the result is: $- 2 \cos{\left(x \right)}$
  2. 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: $3$
  The result is: $3 - 2 \cos{\left(x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \cos{\left(x \right)}$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    The result of the chain rule is:
    
    $- 2 \sin{\left(x \right)} \cos{\left(x \right)}$
  So, the result is: $- 10 \sin{\left(x \right)} \cos{\left(x \right)}$
Now plug in to the quotient rule:

$\frac{5 \left(3 - 2 \cos{\left(x \right)}\right) \cos^{2}{\left(x \right)} + 10 \left(3 x - 2 \sin{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}}{25 \cos^{4}{\left(x \right)}}$
Now simplify:

$\frac{6 x \sin{\left(x \right)} + 3 \cos{\left(x \right)} + \cos{\left(2 x \right)} - 3}{5 \cos^{3}{\left(x \right)}}$

The answer is:

$\frac{6 x \sin{\left(x \right)} + 3 \cos{\left(x \right)} + \cos{\left(2 x \right)} - 3}{5 \cos^{3}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    1                      2*(3*x - 2*sin(x))*sin(x)
---------*(3 - 2*cos(x)) + -------------------------
     2                                  3           
5*cos (x)                          5*cos (x)

$$\left(3 - 2 \cos{\left(x \right)}\right) \frac{1}{5 \cos^{2}{\left(x \right)}} + \frac{2 \left(3 x - 2 \sin{\left(x \right)}\right) \sin{\left(x \right)}}{5 \cos^{3}{\left(x \right)}}$$

Simplify

The second derivative [src]

  //         2   \                                                      \
  ||    3*sin (x)|                     2*(-3 + 2*cos(x))*sin(x)         |
2*||1 + ---------|*(-2*sin(x) + 3*x) - ------------------------ + sin(x)|
  ||        2    |                              cos(x)                  |
  \\     cos (x) /                                                      /
-------------------------------------------------------------------------
                                     2                                   
                                5*cos (x)

$$\frac{2 \left(\left(3 x - 2 \sin{\left(x \right)}\right) \left(\frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) - \frac{2 \left(2 \cos{\left(x \right)} - 3\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + \sin{\left(x \right)}\right)}{5 \cos^{2}{\left(x \right)}}$$

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of (3x-2sinx)/(5cos^2x)

The graph:

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The solution