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Identical expressions
f(x)= seven + one /sin^2x
f(x) equally 7 plus 1 divide by sinus of squared x
f(x) equally seven plus one divide by sinus of squared x
f(x)=7+1/sin2x
fx=7+1/sin2x
f(x)=7+1/sin²x
f(x)=7+1/sin to the power of 2x
fx=7+1/sin^2x
f(x)=7+1 divide by sin^2x
Similar expressions
f(x)=7-1/sin^2x

The derivative of the function/ f(x)=7+1/sin^2x

Derivative of f(x)=7+1/sin^2x

The solution

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       1   
7 + -------
       2   
    sin (x)

$$7 + \frac{1}{\sin^{2}{\left(x \right)}}$$

7 + 1/(sin(x)^2)

Detail solution

Differentiate $7 + \frac{1}{\sin^{2}{\left(x \right)}}$ term by term:
1. The derivative of the constant $7$ is zero.
2. Let $u = \sin^{2}{\left(x \right)}$ .
3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin^{2}{\left(x \right)}$ :
  1. Let $u = \sin{\left(x \right)}$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
    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:
    
    $2 \sin{\left(x \right)} \cos{\left(x \right)}$
  The result of the chain rule is:
  
  $- \frac{2 \cos{\left(x \right)}}{\sin^{3}{\left(x \right)}}$
The result is: $- \frac{2 \cos{\left(x \right)}}{\sin^{3}{\left(x \right)}}$

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Identical expressions

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Derivative of f(x)=7+1/sin^2x

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The solution