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cos(x)^(25)
The derivative of the function/ cos(x)^(25)

Derivative of cos(x)^(25)

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

The graph:

from to

Piecewise:

The solution

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

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

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

  3. Then, apply the chain rule. Multiply by ddxcos⁡(x)\frac{d}{d x} \cos{\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)}

    The result of the chain rule is:

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

The answer is:

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