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cos^34x
The derivative of the function/ cos^34x

Derivative of cos^34x

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

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The solution

You have entered [src] 
   34   
cos  (x)
cos⁡34(x)\cos^{34}{\left(x \right)}
Transform
Detail solution

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

  2. Apply the power rule: u34u^{34} goes to 34u3334 u^{33}

  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:

    −34sin⁡(x)cos⁡33(x)- 34 \sin{\left(x \right)} \cos^{33}{\left(x \right)}

The answer is:

−34sin⁡(x)cos⁡33(x)- 34 \sin{\left(x \right)} \cos^{33}{\left(x \right)}

The graph
02468-8-6-4-2-1010-1010
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
       33          
-34*cos  (x)*sin(x)
−34sin⁡(x)cos⁡33(x)- 34 \sin{\left(x \right)} \cos^{33}{\left(x \right)}
Simplify
The second derivative [src] 
      32    /     2            2   \
34*cos  (x)*\- cos (x) + 33*sin (x)/
34(33sin⁡2(x)−cos⁡2(x))cos⁡32(x)34 \left(33 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos^{32}{\left(x \right)}
Simplify
The third derivative [src] 
       31    /         2            2   \       
136*cos  (x)*\- 264*sin (x) + 25*cos (x)/*sin(x)
136(−264sin⁡2(x)+25cos⁡2(x))sin⁡(x)cos⁡31(x)136 \left(- 264 \sin^{2}{\left(x \right)} + 25 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \cos^{31}{\left(x \right)}
Simplify
The graph
Derivative of cos^34x
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