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lnsin(x^2+1)

Derivative of lnsin(x^2+1)

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
   /   / 2    \\
log\sin\x  + 1//
$$\log{\left(\sin{\left(x^{2} + 1 \right)} \right)}$$
d /   /   / 2    \\\
--\log\sin\x  + 1///
dx                  
$$\frac{d}{d x} \log{\left(\sin{\left(x^{2} + 1 \right)} \right)}$$
Detail solution
  1. Let .

  2. The derivative of is .

  3. Then, apply the chain rule. Multiply by :

    1. Let .

    2. The derivative of sine is cosine:

    3. Then, apply the chain rule. Multiply by :

      1. Differentiate term by term:

        1. Apply the power rule: goes to

        2. The derivative of the constant is zero.

        The result is:

      The result of the chain rule is:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
       / 2    \
2*x*cos\x  + 1/
---------------
     / 2    \  
  sin\x  + 1/  
$$\frac{2 x \cos{\left(x^{2} + 1 \right)}}{\sin{\left(x^{2} + 1 \right)}}$$
The second derivative [src]
  /            /     2\      2    2/     2\\
  |     2   cos\1 + x /   2*x *cos \1 + x /|
2*|- 2*x  + ----------- - -----------------|
  |            /     2\         2/     2\  |
  \         sin\1 + x /      sin \1 + x /  /
$$2 \left(- 2 x^{2} - \frac{2 x^{2} \cos^{2}{\left(x^{2} + 1 \right)}}{\sin^{2}{\left(x^{2} + 1 \right)}} + \frac{\cos{\left(x^{2} + 1 \right)}}{\sin{\left(x^{2} + 1 \right)}}\right)$$
The third derivative [src]
    /          2/     2\      2    3/     2\      2    /     2\\
    |     3*cos \1 + x /   4*x *cos \1 + x /   4*x *cos\1 + x /|
4*x*|-3 - -------------- + ----------------- + ----------------|
    |         2/     2\          3/     2\          /     2\   |
    \      sin \1 + x /       sin \1 + x /       sin\1 + x /   /
$$4 x \left(\frac{4 x^{2} \cos{\left(x^{2} + 1 \right)}}{\sin{\left(x^{2} + 1 \right)}} + \frac{4 x^{2} \cos^{3}{\left(x^{2} + 1 \right)}}{\sin^{3}{\left(x^{2} + 1 \right)}} - 3 - \frac{3 \cos^{2}{\left(x^{2} + 1 \right)}}{\sin^{2}{\left(x^{2} + 1 \right)}}\right)$$
The graph
Derivative of lnsin(x^2+1)