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sin(x)^(56)
The derivative of the function/ sin(x)^(56)

Derivative of sin(x)^(56)

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

You have entered [src] 
   56   
sin  (x)
sin⁡56(x)\sin^{56}{\left(x \right)} 
d /   56   \
--\sin  (x)/
dx
ddxsin⁡56(x)\frac{d}{d x} \sin^{56}{\left(x \right)}
Transform
Detail solution

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

  2. Apply the power rule: u56u^{56} goes to 56u5556 u^{55}

  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:

    56sin⁡55(x)cos⁡(x)56 \sin^{55}{\left(x \right)} \cos{\left(x \right)}

The answer is:

56sin⁡55(x)cos⁡(x)56 \sin^{55}{\left(x \right)} \cos{\left(x \right)}

The graph
02468-8-6-4-2-1010-1010
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
      55          
56*sin  (x)*cos(x)
56sin⁡55(x)cos⁡(x)56 \sin^{55}{\left(x \right)} \cos{\left(x \right)}
Simplify
The second derivative [src] 
      54    /     2            2   \
56*sin  (x)*\- sin (x) + 55*cos (x)/
56(−sin⁡2(x)+55cos⁡2(x))sin⁡54(x)56 \left(- \sin^{2}{\left(x \right)} + 55 \cos^{2}{\left(x \right)}\right) \sin^{54}{\left(x \right)}
Simplify
The third derivative [src] 
       53    /        2              2   \       
112*sin  (x)*\- 83*sin (x) + 1485*cos (x)/*cos(x)
112(−83sin⁡2(x)+1485cos⁡2(x))sin⁡53(x)cos⁡(x)112 \left(- 83 \sin^{2}{\left(x \right)} + 1485 \cos^{2}{\left(x \right)}\right) \sin^{53}{\left(x \right)} \cos{\left(x \right)}
Simplify
The graph
Derivative of sin(x)^(56)
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