Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of (x-3)^4
Derivative of (x+3)^3
Derivative of (x^2-1)^3
Derivative of (x+1)/(x^2+1)
Identical expressions
two *cos^ two (x)-(sin(2x))
2 multiply by co sinus of e of squared (x) minus ( sinus of (2x))
two multiply by co sinus of e of to the power of two (x) minus ( sinus of (2x))
2*cos2(x)-(sin(2x))
2*cos2x-sin2x
2*cos²(x)-(sin(2x))
2*cos to the power of 2(x)-(sin(2x))
2cos^2(x)-(sin(2x))
2cos2(x)-(sin(2x))
2cos2x-sin2x
2cos^2x-sin2x
Similar expressions
2*cos^2(x)+(sin(2x))

The derivative of the function/ 2*cos^2(x)-(sin(2x))

Derivative of 2*cos^2(x)-(sin(2x))

The solution

You have entered [src]

     2              
2*cos (x) - sin(2*x)

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

2*cos(x)^2 - sin(2*x)

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /   2         2              \
4*\sin (x) - cos (x) + sin(2*x)/

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

Simplify

The third derivative [src]

8*(2*cos(x)*sin(x) + cos(2*x))

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

Simplify