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y=sin one /x+1/sinx
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Similar expressions
y=sin1/x-1/sinx

The derivative of the function/ y=sin1/x+1/sinx

Derivative of y=sin1/x+1/sinx

The solution

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sin(1)     1   
------ + ------
  x      sin(x)

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

sin(1)/x + 1/sin(x)

Detail solution

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  sin(1)    cos(x)
- ------ - -------
     2        2   
    x      sin (x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of y=sin1/x+1/sinx

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The solution