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sin^23x
The derivative of the function/ sin^23x

Derivative of sin^23x

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

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The solution

You have entered [src] 
   23   
sin  (x)
sin⁡23(x)\sin^{23}{\left(x \right)} 
sin(x)^23
Detail solution

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

  2. Apply the power rule: u23u^{23} goes to 23u2223 u^{22}

  3. 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:

    23sin⁡22(x)cos⁡(x)23 \sin^{22}{\left(x \right)} \cos{\left(x \right)}

The answer is:

23sin⁡22(x)cos⁡(x)23 \sin^{22}{\left(x \right)} \cos{\left(x \right)}

The graph
02468-8-6-4-2-10105-5
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
      22          
23*sin  (x)*cos(x)
23sin⁡22(x)cos⁡(x)23 \sin^{22}{\left(x \right)} \cos{\left(x \right)}
Simplify
The second derivative [src] 
      21    /     2            2   \
23*sin  (x)*\- sin (x) + 22*cos (x)/
23(−sin⁡2(x)+22cos⁡2(x))sin⁡21(x)23 \left(- \sin^{2}{\left(x \right)} + 22 \cos^{2}{\left(x \right)}\right) \sin^{21}{\left(x \right)}
Simplify
The third derivative [src] 
      20    /        2             2   \       
23*sin  (x)*\- 67*sin (x) + 462*cos (x)/*cos(x)
23(−67sin⁡2(x)+462cos⁡2(x))sin⁡20(x)cos⁡(x)23 \left(- 67 \sin^{2}{\left(x \right)} + 462 \cos^{2}{\left(x \right)}\right) \sin^{20}{\left(x \right)} \cos{\left(x \right)}
Simplify
The graph
Derivative of sin^23x
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