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Derivative of:
Derivative of (-2)/x^2
Derivative of (cos(x))^sin(x)
Derivative of y=(x+2)²(2x-1)³
Derivative of y=5x-6
Identical expressions
y=lnsin(e^x^ two)
y equally ln sinus of (e to the power of x squared )
y equally ln sinus of (e to the power of x to the power of two)
y=lnsin(ex2)
y=lnsinex2
y=lnsin(e^x²)
y=lnsin(e to the power of x to the power of 2)
y=lnsine^x^2

The derivative of the function/ y=lnsin(e^x^2)

Derivative of y=lnsin(e^x^2)

The solution

You have entered [src]

   /   / / 2\\\
   |   | \x /||
log\sin\e    //

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

  /   /   / / 2\\\\
d |   |   | \x /|||
--\log\sin\e    ///
dx

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

Transform

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       / / 2\\  / 2\
       | \x /|  \x /
2*x*cos\e    /*e    
--------------------
        / / 2\\     
        | \x /|     
     sin\e    /

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

Simplify

The second derivative [src]

  /   / / 2\\                        / / 2\\            / / 2\\  / 2\\      
  |   | \x /|         / 2\      2    | \x /|      2    2| \x /|  \x /|  / 2\
  |cos\e    /      2  \x /   2*x *cos\e    /   2*x *cos \e    /*e    |  \x /
2*|---------- - 2*x *e     + --------------- - ----------------------|*e    
  |   / / 2\\                      / / 2\\              / / 2\\      |      
  |   | \x /|                      | \x /|             2| \x /|      |      
  \sin\e    /                   sin\e    /          sin \e    /      /

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

Simplify

The third derivative [src]

    /                              / / 2\\         / / 2\\  / 2\           / / 2\\            / / 2\\  / 2\            / / 2\\     2           / / 2\\     2\      
    |     / 2\         / 2\        | \x /|        2| \x /|  \x /      2    | \x /|      2    2| \x /|  \x /      2    3| \x /|  2*x       2    | \x /|  2*x |  / 2\
    |     \x /      2  \x /   3*cos\e    /   3*cos \e    /*e       2*x *cos\e    /   6*x *cos \e    /*e       4*x *cos \e    /*e       4*x *cos\e    /*e    |  \x /
4*x*|- 3*e     - 6*x *e     + ------------ - ------------------- + --------------- - ---------------------- + ---------------------- + ---------------------|*e    
    |                             / / 2\\            / / 2\\             / / 2\\              / / 2\\                  / / 2\\                  / / 2\\     |      
    |                             | \x /|           2| \x /|             | \x /|             2| \x /|                 3| \x /|                  | \x /|     |      
    \                          sin\e    /        sin \e    /          sin\e    /          sin \e    /              sin \e    /               sin\e    /     /

$$4 x \left(\frac{4 x^{2} e^{2 x^{2}} \cos{\left(e^{x^{2}} \right)}}{\sin{\left(e^{x^{2}} \right)}} + \frac{4 x^{2} e^{2 x^{2}} \cos^{3}{\left(e^{x^{2}} \right)}}{\sin^{3}{\left(e^{x^{2}} \right)}} - 6 x^{2} e^{x^{2}} - \frac{6 x^{2} e^{x^{2}} \cos^{2}{\left(e^{x^{2}} \right)}}{\sin^{2}{\left(e^{x^{2}} \right)}} + \frac{2 x^{2} \cos{\left(e^{x^{2}} \right)}}{\sin{\left(e^{x^{2}} \right)}} - 3 e^{x^{2}} - \frac{3 e^{x^{2}} \cos^{2}{\left(e^{x^{2}} \right)}}{\sin^{2}{\left(e^{x^{2}} \right)}} + \frac{3 \cos{\left(e^{x^{2}} \right)}}{\sin{\left(e^{x^{2}} \right)}}\right) e^{x^{2}}$$

Simplify

The graph

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Identical expressions

Derivative of y=lnsin(e^x^2)

The graph:

Piecewise:

The solution