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cosx^3
The derivative of the function/ cosx^3

Derivative of cosx^3

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

The graph:

from to

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The solution

You have entered [src] 
   3   
cos (x)
cos⁡3(x)\cos^{3}{\left(x \right)} 
d /   3   \
--\cos (x)/
dx
ddxcos⁡3(x)\frac{d}{d x} \cos^{3}{\left(x \right)}
Transform
Detail solution

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

  2. Apply the power rule: u3u^{3} goes to 3u23 u^{2}

  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:

    −3sin⁡(x)cos⁡2(x)- 3 \sin{\left(x \right)} \cos^{2}{\left(x \right)}

The answer is:

−3sin⁡(x)cos⁡2(x)- 3 \sin{\left(x \right)} \cos^{2}{\left(x \right)}

The graph
02468-8-6-4-2-10102.5-2.5
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
      2          
-3*cos (x)*sin(x)
−3sin⁡(x)cos⁡2(x)- 3 \sin{\left(x \right)} \cos^{2}{\left(x \right)}
Simplify
The second derivative [src] 
  /     2           2   \       
3*\- cos (x) + 2*sin (x)/*cos(x)
3⋅(2sin⁡2(x)−cos⁡2(x))cos⁡(x)3 \cdot \left(2 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos{\left(x \right)}
Simplify
The third derivative [src] 
  /       2           2   \       
3*\- 2*sin (x) + 7*cos (x)/*sin(x)
3(−2sin⁡2(x)+7cos⁡2(x))sin⁡(x)3 \left(- 2 \sin^{2}{\left(x \right)} + 7 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}
Simplify
The graph
Derivative of cosx^3
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