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csc(x)
The derivative of the function/ csc(x)

Derivative of csc(x)

Function f() - derivative -N order at the point
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The graph:

from to

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The solution

You have entered [src] 
csc(x)
csc⁡(x)\csc{\left(x \right)} 
csc(x)
Detail solution

  1. Rewrite the function to be differentiated:

    csc⁡(x)=1sin⁡(x)\csc{\left(x \right)} = \frac{1}{\sin{\left(x \right)}}

  2. Let u=sin⁡(x)u = \sin{\left(x \right)}.

  3. Apply the power rule: 1u\frac{1}{u} goes to −1u2- \frac{1}{u^{2}}

  4. Then, apply the chain rule. Multiply by ddxsin⁡(x)\frac{d}{d x} \sin{\left(x \right)}:

    1. The derivative of sine is cosine:

      ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

    The result of the chain rule is:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-1000010000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
-cot(x)*csc(x)
−cot⁡(x)csc⁡(x)- \cot{\left(x \right)} \csc{\left(x \right)}
Simplify
The second derivative [src] 
/         2   \       
\1 + 2*cot (x)/*csc(x)
(2cot⁡2(x)+1)csc⁡(x)\left(2 \cot^{2}{\left(x \right)} + 1\right) \csc{\left(x \right)}
Simplify
The third derivative [src] 
 /         2   \              
-\5 + 6*cot (x)/*cot(x)*csc(x)
−(6cot⁡2(x)+5)cot⁡(x)csc⁡(x)- \left(6 \cot^{2}{\left(x \right)} + 5\right) \cot{\left(x \right)} \csc{\left(x \right)}
Simplify
The graph
Derivative of csc(x)
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