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cos(x)^(42)
The derivative of the function/ cos(x)^(42)

Derivative of cos(x)^(42)

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   42   
cos  (x)
$$\cos^{42}{\left(x \right)}$$ 
cos(x)^42
Detail solution

  1. Let .

  2. Apply the power rule: goes to

  3. Then, apply the chain rule. Multiply by :

    1. The derivative of cosine is negative sine:

    The result of the chain rule is:

The answer is:

The graph
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
       41          
-42*cos  (x)*sin(x)
$$- 42 \sin{\left(x \right)} \cos^{41}{\left(x \right)}$$
Simplify
The second derivative [src] 
      40    /     2            2   \
42*cos  (x)*\- cos (x) + 41*sin (x)/
$$42 \left(41 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos^{40}{\left(x \right)}$$
Simplify
The third derivative [src] 
       39    /         2            2   \       
168*cos  (x)*\- 410*sin (x) + 31*cos (x)/*sin(x)
$$168 \left(- 410 \sin^{2}{\left(x \right)} + 31 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \cos^{39}{\left(x \right)}$$
Simplify
The graph
Derivative of cos(x)^(42)
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