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Derivative of:
Derivative of x^12
Derivative of -e^x
Derivative of (1+x)/(4-x^2)
Derivative of (1+sqrt(x))/(1-sqrt(x))
Identical expressions
five +4cos^ two (x)
5 plus 4 co sinus of e of squared (x)
five plus 4 co sinus of e of to the power of two (x)
5+4cos2(x)
5+4cos2x
5+4cos²(x)
5+4cos to the power of 2(x)
5+4cos^2x
Similar expressions
5-4cos^2(x)

The derivative of the function/ 5+4cos^2(x)

Derivative of 5+4cos^2(x)

The solution

You have entered [src]

         2   
5 + 4*cos (x)

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

5 + 4*cos(x)^2

Detail solution

Differentiate $4 \cos^{2}{\left(x \right)} + 5$ term by term:
1. The derivative of the constant $5$ is zero.
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \cos{\left(x \right)}$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    The result of the chain rule is:
    
    $- 2 \sin{\left(x \right)} \cos{\left(x \right)}$
  So, the result is: $- 8 \sin{\left(x \right)} \cos{\left(x \right)}$
The result is: $- 8 \sin{\left(x \right)} \cos{\left(x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

-8*cos(x)*sin(x)

$$- 8 \sin{\left(x \right)} \cos{\left(x \right)}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

32*cos(x)*sin(x)

$$32 \sin{\left(x \right)} \cos{\left(x \right)}$$

Simplify