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y=sinxcosxcscx
The derivative of the function/ y=sinxcosxcscx

Derivative of y=sinxcosxcscx

Function f() - derivative -N order at the point
v

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The solution

You have entered [src] 
sin(x)*cos(x)*csc(x)
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
ddxsin⁡(x)cos⁡(x)csc⁡(x)\frac{d}{d x} \sin{\left(x \right)} \cos{\left(x \right)} \csc{\left(x \right)}
Transform
Detail solution

  1. Apply the product rule:

    ddxf(x)g(x)h(x)=f(x)g(x)ddxh(x)+f(x)h(x)ddxg(x)+g(x)h(x)ddxf(x)\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(x)=sin⁡(x)f{\left(x \right)} = \sin{\left(x \right)}; to find ddxf(x)\frac{d}{d x} f{\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)}

    g(x)=cos⁡(x)g{\left(x \right)} = \cos{\left(x \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. The derivative of cosine is negative sine:

      ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

    h(x)=csc⁡(x)h{\left(x \right)} = \csc{\left(x \right)}; to find ddxh(x)\frac{d}{d x} h{\left(x \right)}:

    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 result is: −sin⁡2(x)csc⁡(x)+cos⁡2(x)csc⁡(x)−cos⁡2(x)sin⁡(x)- \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)}}

  2. Now simplify:

    −sin⁡(x)- \sin{\left(x \right)}

The answer is:

−sin⁡(x)- \sin{\left(x \right)}

The graph
02468-8-6-4-2-10102-2
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⁡(x)cos⁡(x)cot⁡(x)csc⁡(x)−sin⁡2(x)csc⁡(x)+cos⁡2(x)csc⁡(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)
((2cot⁡2(x)+1)sin⁡(x)cos⁡(x)+2sin⁡2(x)cot⁡(x)−2cos⁡2(x)cot⁡(x)−4sin⁡(x)cos⁡(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)
(−(6cot⁡2(x)+5)sin⁡(x)cos⁡(x)cot⁡(x)−3⋅(2cot⁡2(x)+1)sin⁡2(x)+3⋅(2cot⁡2(x)+1)cos⁡2(x)+12sin⁡(x)cos⁡(x)cot⁡(x)+4sin⁡2(x)−4cos⁡2(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
Derivative of y=sinxcosxcscx
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