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Derivative of:
Derivative of x^(4/5)
Derivative of x^2+5
Derivative of x^3-1/5*x^2+2*x-4
Derivative of |x-1|
Integral of d{x}:
cos^2(2t)
Identical expressions
cos^ two (2t)
co sinus of e of squared (2t)
co sinus of e of to the power of two (2t)
cos2(2t)
cos22t
cos²(2t)
cos to the power of 2(2t)
cos^22t

The derivative of the function/ cos^2(2t)

Derivative of cos^2(2t)

The solution

You have entered [src]

   2     
cos (2*t)

$$\cos^{2}{\left(2 t \right)}$$

d /   2     \
--\cos (2*t)/
dt

$$\frac{d}{d t} \cos^{2}{\left(2 t \right)}$$

Transform

Detail solution

Let $u = \cos{\left(2 t \right)}$ .
Apply the power rule: $u^{2}$ goes to $2 u$
Then, apply the chain rule. Multiply by $\frac{d}{d t} \cos{\left(2 t \right)}$ :
1. Let $u = 2 t$ .
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 $\frac{d}{d t} 2 t$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $t$ goes to $1$
    So, the result is: $2$
  The result of the chain rule is:
  
  $- 2 \sin{\left(2 t \right)}$
The result of the chain rule is:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

-4*cos(2*t)*sin(2*t)

$$- 4 \sin{\left(2 t \right)} \cos{\left(2 t \right)}$$

Simplify

The second derivative [src]

  /   2           2     \
8*\sin (2*t) - cos (2*t)/

$$8 \left(\sin^{2}{\left(2 t \right)} - \cos^{2}{\left(2 t \right)}\right)$$

Simplify

The third derivative [src]

64*cos(2*t)*sin(2*t)

$$64 \sin{\left(2 t \right)} \cos{\left(2 t \right)}$$

Simplify

The graph

Derivative of cos^2(2t)