Mister Exam

Lang:

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

You have entered [src]

              2
(sin(2*x) + 1)

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

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

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

Transform

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

-16*(1 + 4*sin(2*x))*cos(2*x)

$$- 16 \cdot \left(4 \sin{\left(2 x \right)} + 1\right) \cos{\left(2 x \right)}$$

Simplify

The graph