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sin²5x
The derivative of the function/ sin²5x

Derivative of sin²5x

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   25   
sin  (x)
sin⁡25(x)\sin^{25}{\left(x \right)} 
d /   25   \
--\sin  (x)/
dx
ddxsin⁡25(x)\frac{d}{d x} \sin^{25}{\left(x \right)}
Transform
Detail solution

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

  2. Apply the power rule: u25u^{25} goes to 25u2425 u^{24}

  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:

    25sin⁡24(x)cos⁡(x)25 \sin^{24}{\left(x \right)} \cos{\left(x \right)}

The answer is:

25sin⁡24(x)cos⁡(x)25 \sin^{24}{\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] 
      24          
25*sin  (x)*cos(x)
25sin⁡24(x)cos⁡(x)25 \sin^{24}{\left(x \right)} \cos{\left(x \right)}
Simplify
The second derivative [src] 
      23    /     2            2   \
25*sin  (x)*\- sin (x) + 24*cos (x)/
25(−sin⁡2(x)+24cos⁡2(x))sin⁡23(x)25 \left(- \sin^{2}{\left(x \right)} + 24 \cos^{2}{\left(x \right)}\right) \sin^{23}{\left(x \right)}
Simplify
The third derivative [src] 
      22    /        2             2   \       
25*sin  (x)*\- 73*sin (x) + 552*cos (x)/*cos(x)
25(−73sin⁡2(x)+552cos⁡2(x))sin⁡22(x)cos⁡(x)25 \left(- 73 \sin^{2}{\left(x \right)} + 552 \cos^{2}{\left(x \right)}\right) \sin^{22}{\left(x \right)} \cos{\left(x \right)}
Simplify
The graph
Derivative of sin²5x
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