Derivative of y=sinxcosxcscx

The solution

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sin(x)*cos(x)*csc(x)

$$\sin{\left(x \right)} \cos{\left(x \right)} \csc{\left(x \right)}$$

d                       
--(sin(x)*cos(x)*csc(x))
dx

$$\frac{d}{d x} \sin{\left(x \right)} \cos{\left(x \right)} \csc{\left(x \right)}$$

Transform

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} h{\left(x \right)} = f{\left(x \right)} g{\left(x \right)} \frac{d}{d x} h{\left(x \right)} + f{\left(x \right)} h{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} h{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
$g{\left(x \right)} = \cos{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of cosine is negative sine:
  
  $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
$h{\left(x \right)} = \csc{\left(x \right)}$ ; to find $\frac{d}{d x} h{\left(x \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\csc{\left(x \right)} = \frac{1}{\sin{\left(x \right)}}$
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: $- \sin^{2}{\left(x \right)} \csc{\left(x \right)} + \cos^{2}{\left(x \right)} \csc{\left(x \right)} - \frac{\cos^{2}{\left(x \right)}}{\sin{\left(x \right)}}$
Now simplify:

$- \sin{\left(x \right)}$

The answer is:

$- \sin{\left(x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

$$- \sin{\left(x \right)} \cos{\left(x \right)} \cot{\left(x \right)} \csc{\left(x \right)} - \sin^{2}{\left(x \right)} \csc{\left(x \right)} + \cos^{2}{\left(x \right)} \csc{\left(x \right)}$$

Simplify

The second derivative [src]

/                        2                  2             /         2   \              \       
\-4*cos(x)*sin(x) - 2*cos (x)*cot(x) + 2*sin (x)*cot(x) + \1 + 2*cot (x)/*cos(x)*sin(x)/*csc(x)

$$\left(\left(2 \cot^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)} + 2 \sin^{2}{\left(x \right)} \cot{\left(x \right)} - 2 \cos^{2}{\left(x \right)} \cot{\left(x \right)} - 4 \sin{\left(x \right)} \cos{\left(x \right)}\right) \csc{\left(x \right)}$$

Simplify

The third derivative [src]

/       2           2           2    /         2   \        2    /         2   \                             /         2   \                     \       
\- 4*cos (x) + 4*sin (x) - 3*sin (x)*\1 + 2*cot (x)/ + 3*cos (x)*\1 + 2*cot (x)/ + 12*cos(x)*cot(x)*sin(x) - \5 + 6*cot (x)/*cos(x)*cot(x)*sin(x)/*csc(x)

$$\left(- \left(6 \cot^{2}{\left(x \right)} + 5\right) \sin{\left(x \right)} \cos{\left(x \right)} \cot{\left(x \right)} - 3 \cdot \left(2 \cot^{2}{\left(x \right)} + 1\right) \sin^{2}{\left(x \right)} + 3 \cdot \left(2 \cot^{2}{\left(x \right)} + 1\right) \cos^{2}{\left(x \right)} + 12 \sin{\left(x \right)} \cos{\left(x \right)} \cot{\left(x \right)} + 4 \sin^{2}{\left(x \right)} - 4 \cos^{2}{\left(x \right)}\right) \csc{\left(x \right)}$$

Simplify

The graph

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Derivative of y=sinxcosxcscx

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The solution