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Derivative of cos^4(2x+5)

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   4         
cos (2*x + 5)

$$\cos^{4}{\left(2 x + 5 \right)}$$

cos(2*x + 5)^4

Detail solution

Let $u = \cos{\left(2 x + 5 \right)}$ .
Apply the power rule: $u^{4}$ goes to $4 u^{3}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(2 x + 5 \right)}$ :
1. Let $u = 2 x + 5$ .
2. The derivative of cosine is negative sine:
  
  $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x + 5\right)$ :
  1. Differentiate $2 x + 5$ term by term:
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $2$
    2. The derivative of the constant $5$ is zero.
    The result is: $2$
  The result of the chain rule is:
  
  $- 2 \sin{\left(2 x + 5 \right)}$
The result of the chain rule is:

$- 8 \sin{\left(2 x + 5 \right)} \cos^{3}{\left(2 x + 5 \right)}$
Now simplify:

$- 8 \sin{\left(2 x + 5 \right)} \cos^{3}{\left(2 x + 5 \right)}$

The answer is:

$- 8 \sin{\left(2 x + 5 \right)} \cos^{3}{\left(2 x + 5 \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      3                      
-8*cos (2*x + 5)*sin(2*x + 5)

$$- 8 \sin{\left(2 x + 5 \right)} \cos^{3}{\left(2 x + 5 \right)}$$

Simplify

The second derivative [src]

      2          /     2                 2         \
16*cos (5 + 2*x)*\- cos (5 + 2*x) + 3*sin (5 + 2*x)/

$$16 \left(3 \sin^{2}{\left(2 x + 5 \right)} - \cos^{2}{\left(2 x + 5 \right)}\right) \cos^{2}{\left(2 x + 5 \right)}$$

Simplify

The third derivative [src]

   /       2                 2         \                          
64*\- 3*sin (5 + 2*x) + 5*cos (5 + 2*x)/*cos(5 + 2*x)*sin(5 + 2*x)

$$64 \left(- 3 \sin^{2}{\left(2 x + 5 \right)} + 5 \cos^{2}{\left(2 x + 5 \right)}\right) \sin{\left(2 x + 5 \right)} \cos{\left(2 x + 5 \right)}$$

Simplify