Derivative of csc7x

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csc(7*x)

$$\csc{\left(7 x \right)}$$

d           
--(csc(7*x))
dx

$$\frac{d}{d x} \csc{\left(7 x \right)}$$

Transform

Detail solution

Rewrite the function to be differentiated:

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

$- \frac{7 \cos{\left(7 x \right)}}{\sin^{2}{\left(7 x \right)}}$

The answer is:

$- \frac{7 \cos{\left(7 x \right)}}{\sin^{2}{\left(7 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

-7*cot(7*x)*csc(7*x)

$$- 7 \cot{\left(7 x \right)} \csc{\left(7 x \right)}$$

Simplify

The second derivative [src]

   /         2     \         
49*\1 + 2*cot (7*x)/*csc(7*x)

$$49 \cdot \left(2 \cot^{2}{\left(7 x \right)} + 1\right) \csc{\left(7 x \right)}$$

Simplify

The third derivative [src]

     /         2     \                  
-343*\5 + 6*cot (7*x)/*cot(7*x)*csc(7*x)

$$- 343 \cdot \left(6 \cot^{2}{\left(7 x \right)} + 5\right) \cot{\left(7 x \right)} \csc{\left(7 x \right)}$$

Simplify

The graph