Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of cos(x/3)
Derivative of √x
Derivative of xcos(x)
Derivative of ln1
Identical expressions
ln(one +sin^ two (x))
ln(1 plus sinus of squared (x))
ln(one plus sinus of to the power of two (x))
ln(1+sin2(x))
ln1+sin2x
ln(1+sin²(x))
ln(1+sin to the power of 2(x))
ln1+sin^2x
Similar expressions
ln(1-sin^2(x))

The derivative of the function/ ln(1+sin^2(x))

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

The solution

You have entered [src]

   /       2   \
log\1 + sin (x)/

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

log(1 + sin(x)^2)

Detail solution

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

$\frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1}$
Now simplify:

$\frac{2 \sin{\left(2 x \right)}}{3 - \cos{\left(2 x \right)}}$

The answer is:

$\frac{2 \sin{\left(2 x \right)}}{3 - \cos{\left(2 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

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

The graph:

Piecewise:

The solution