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Similar expressions
(1-sin(2x))*cos(2x)

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

Derivative of (1+sin(2x))*cos(2x)

The solution

You have entered [src]

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

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

d                          
--((1 + sin(2*x))*cos(2*x))
dx

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

Transform

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = \sin{\left(2 x \right)} + 1$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $\sin{\left(2 x \right)} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Let $u = 2 x$ .
  3. The derivative of sine is cosine:
    
    $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
  4. 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:
    
    $2 \cos{\left(2 x \right)}$
  The result is: $2 \cos{\left(2 x \right)}$
$g{\left(x \right)} = \cos{\left(2 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 2 x$ .
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 $\frac{d}{d x} 2 x$ :
  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:
  
  $- 2 \sin{\left(2 x \right)}$
The result is: $- 2 \left(\sin{\left(2 x \right)} + 1\right) \sin{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}$

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /       2             2                               \
8*\- 4*cos (2*x) + 3*sin (2*x) + (1 + sin(2*x))*sin(2*x)/

$$8 \left(\left(\sin{\left(2 x \right)} + 1\right) \sin{\left(2 x \right)} + 3 \sin^{2}{\left(2 x \right)} - 4 \cos^{2}{\left(2 x \right)}\right)$$

Simplify

The graph

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Derivative of (1+sin(2x))*cos(2x)

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The solution