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Identical expressions
cos^ two (2x+ one)
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Similar expressions
cos^2(2x-1)

The derivative of the function/ cos^2(2x+1)

Derivative of cos^2(2x+1)

The solution

You have entered [src]

   2         
cos (2*x + 1)

$$\cos^{2}{\left(2 x + 1 \right)}$$

d /   2         \
--\cos (2*x + 1)/
dx

$$\frac{d}{d x} \cos^{2}{\left(2 x + 1 \right)}$$

Transform

Detail solution

Let $u = \cos{\left(2 x + 1 \right)}$ .
Apply the power rule: $u^{2}$ goes to $2 u$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(2 x + 1 \right)}$ :
1. Let $u = 2 x + 1$ .
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 $\frac{d}{d x} \left(2 x + 1\right)$ :
  1. Differentiate $2 x + 1$ 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 $1$ is zero.
    The result is: $2$
  The result of the chain rule is:
  
  $- 2 \sin{\left(2 x + 1 \right)}$
The result of the chain rule is:

$- 4 \sin{\left(2 x + 1 \right)} \cos{\left(2 x + 1 \right)}$
Now simplify:

$- 2 \sin{\left(4 x + 2 \right)}$

The answer is:

$- 2 \sin{\left(4 x + 2 \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

-4*cos(2*x + 1)*sin(2*x + 1)

$$- 4 \sin{\left(2 x + 1 \right)} \cos{\left(2 x + 1 \right)}$$

Simplify

The second derivative [src]

  /   2               2         \
8*\sin (1 + 2*x) - cos (1 + 2*x)/

$$8 \left(\sin^{2}{\left(2 x + 1 \right)} - \cos^{2}{\left(2 x + 1 \right)}\right)$$

Simplify

The third derivative [src]

64*cos(1 + 2*x)*sin(1 + 2*x)

$$64 \sin{\left(2 x + 1 \right)} \cos{\left(2 x + 1 \right)}$$

Simplify

The graph

Derivative of cos^2(2x+1)