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Derivative of 4/x^2
Identical expressions
(- nine /sin(x))-(four /cos(x))
( minus 9 divide by sinus of (x)) minus (4 divide by co sinus of e of (x))
( minus nine divide by sinus of (x)) minus (four divide by co sinus of e of (x))
-9/sinx-4/cosx
(-9 divide by sin(x))-(4 divide by cos(x))
Similar expressions
(-9/sin(x))+(4/cos(x))
(9/sin(x))-(4/cos(x))
(-9/sinx)-(4/cosx)

The derivative of the function/ (-9/sin(x))-(4/cos(x))

Derivative of (-9/sin(x))-(4/cos(x))

The solution

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    9        4   
- ------ - ------
  sin(x)   cos(x)

$$- \frac{4}{\cos{\left(x \right)}} - \frac{9}{\sin{\left(x \right)}}$$

-9/sin(x) - 4/cos(x)

Detail solution

Differentiate $- \frac{4}{\cos{\left(x \right)}} - \frac{9}{\sin{\left(x \right)}}$ term by term:
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \sin{\left(x \right)}$ .
  2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  3. 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:
    
    $- \frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
  So, the result is: $\frac{9 \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
2. 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: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  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:
    
    $\frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  So, the result is: $- \frac{4 \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
The result is: $- \frac{4 \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{9 \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}$

The answer is:

$- \frac{4 \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{9 \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  4*sin(x)   9*cos(x)
- -------- + --------
     2          2    
  cos (x)    sin (x)

$$- \frac{4 \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{9 \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}$$

Simplify

The second derivative [src]

 /                       2            2   \
 |  4        9      8*sin (x)   18*cos (x)|
-|------ + ------ + --------- + ----------|
 |cos(x)   sin(x)       3           3     |
 \                   cos (x)     sin (x)  /

$$- (\frac{8 \sin^{2}{\left(x \right)}}{\cos^{3}{\left(x \right)}} + \frac{4}{\cos{\left(x \right)}} + \frac{9}{\sin{\left(x \right)}} + \frac{18 \cos^{2}{\left(x \right)}}{\sin^{3}{\left(x \right)}})$$

Simplify

The third derivative [src]

        3                                    3   
  24*sin (x)   20*sin(x)   45*cos(x)   54*cos (x)
- ---------- - --------- + --------- + ----------
      4            2           2           4     
   cos (x)      cos (x)     sin (x)     sin (x)

$$- \frac{24 \sin^{3}{\left(x \right)}}{\cos^{4}{\left(x \right)}} - \frac{20 \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{45 \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{54 \cos^{3}{\left(x \right)}}{\sin^{4}{\left(x \right)}}$$

Simplify

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Derivative of (-9/sin(x))-(4/cos(x))

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The solution