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cos^6(x)
The derivative of the function/ cos^6(x)

Derivative of cos^6(x)

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

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Piecewise:

The solution

You have entered [src] 
   6   
cos (x)
cos⁡6(x)\cos^{6}{\left(x \right)} 
cos(x)^6
Detail solution

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

  2. Apply the power rule: u6u^{6} goes to 6u56 u^{5}

  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:

    −6sin⁡(x)cos⁡5(x)- 6 \sin{\left(x \right)} \cos^{5}{\left(x \right)}

The answer is:

−6sin⁡(x)cos⁡5(x)- 6 \sin{\left(x \right)} \cos^{5}{\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] 
      5          
-6*cos (x)*sin(x)
−6sin⁡(x)cos⁡5(x)- 6 \sin{\left(x \right)} \cos^{5}{\left(x \right)}
Simplify
The second derivative [src] 
     4    /     2           2   \
6*cos (x)*\- cos (x) + 5*sin (x)/
6(5sin⁡2(x)−cos⁡2(x))cos⁡4(x)6 \left(5 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos^{4}{\left(x \right)}
Simplify
The third derivative [src] 
      3    /       2           2   \       
24*cos (x)*\- 5*sin (x) + 4*cos (x)/*sin(x)
24(−5sin⁡2(x)+4cos⁡2(x))sin⁡(x)cos⁡3(x)24 \left(- 5 \sin^{2}{\left(x \right)} + 4 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \cos^{3}{\left(x \right)}
Simplify
The graph
Derivative of cos^6(x)
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