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Identical expressions
lnsin(x^ two + one)
ln sinus of (x squared plus 1)
ln sinus of (x to the power of two plus one)
lnsin(x2+1)
lnsinx2+1
lnsin(x²+1)
lnsin(x to the power of 2+1)
lnsinx^2+1
Similar expressions
lnsin(x^2-1)

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

Derivative of lnsin(x^2+1)

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

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

Transform

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       / 2    \
2*x*cos\x  + 1/
---------------
     / 2    \  
  sin\x  + 1/

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

Simplify

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 /  /

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

Simplify

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 /   /

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

Simplify

The graph

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Derivative of lnsin(x^2+1)

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Piecewise:

The solution