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cos^4(x)
The derivative of the function/ cos^4(x)

Derivative of cos^4(x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   4   
cos (x)
cos⁡4(x)\cos^{4}{\left(x \right)} 
d /   4   \
--\cos (x)/
dx
ddxcos⁡4(x)\frac{d}{d x} \cos^{4}{\left(x \right)}
Transform
Detail solution

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

  2. Apply the power rule: u4u^{4} goes to 4u34 u^{3}

  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:

    −4sin⁡(x)cos⁡3(x)- 4 \sin{\left(x \right)} \cos^{3}{\left(x \right)}

The answer is:

−4sin⁡(x)cos⁡3(x)- 4 \sin{\left(x \right)} \cos^{3}{\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] 
      3          
-4*cos (x)*sin(x)
−4sin⁡(x)cos⁡3(x)- 4 \sin{\left(x \right)} \cos^{3}{\left(x \right)}
Simplify
The second derivative [src] 
     2    /     2           2   \
4*cos (x)*\- cos (x) + 3*sin (x)/
4⋅(3sin⁡2(x)−cos⁡2(x))cos⁡2(x)4 \cdot \left(3 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos^{2}{\left(x \right)}
Simplify
The third derivative [src] 
  /       2           2   \              
8*\- 3*sin (x) + 5*cos (x)/*cos(x)*sin(x)
8(−3sin⁡2(x)+5cos⁡2(x))sin⁡(x)cos⁡(x)8 \left(- 3 \sin^{2}{\left(x \right)} + 5 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}
Simplify
5-th derivative [src] 
   /        2            2   \              
32*\- 17*cos (x) + 15*sin (x)/*cos(x)*sin(x)
32⋅(15sin⁡2(x)−17cos⁡2(x))sin⁡(x)cos⁡(x)32 \cdot \left(15 \sin^{2}{\left(x \right)} - 17 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}
Simplify
The graph
Derivative of cos^4(x)
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