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cos^7(x)
The derivative of the function/ cos^7(x)

Derivative of cos^7(x)

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

The graph:

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The solution

You have entered [src] 
   7   
cos (x)
cos⁡7(x)\cos^{7}{\left(x \right)} 
d /   7   \
--\cos (x)/
dx
ddxcos⁡7(x)\frac{d}{d x} \cos^{7}{\left(x \right)}
Transform
Detail solution

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

  2. Apply the power rule: u7u^{7} goes to 7u67 u^{6}

  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:

    −7sin⁡(x)cos⁡6(x)- 7 \sin{\left(x \right)} \cos^{6}{\left(x \right)}

The answer is:

−7sin⁡(x)cos⁡6(x)- 7 \sin{\left(x \right)} \cos^{6}{\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] 
      6          
-7*cos (x)*sin(x)
−7sin⁡(x)cos⁡6(x)- 7 \sin{\left(x \right)} \cos^{6}{\left(x \right)}
Simplify
The second derivative [src] 
     5    /     2           2   \
7*cos (x)*\- cos (x) + 6*sin (x)/
7⋅(6sin⁡2(x)−cos⁡2(x))cos⁡5(x)7 \cdot \left(6 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos^{5}{\left(x \right)}
Simplify
The third derivative [src] 
     4    /        2            2   \       
7*cos (x)*\- 30*sin (x) + 19*cos (x)/*sin(x)
7(−30sin⁡2(x)+19cos⁡2(x))sin⁡(x)cos⁡4(x)7 \left(- 30 \sin^{2}{\left(x \right)} + 19 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \cos^{4}{\left(x \right)}
Simplify
The graph
Derivative of cos^7(x)
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