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Derivative of:
Derivative of x+1
Derivative of e^(x^2)
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Identical expressions
y=ln(e^2x+ one)-2arctge^x
y equally ln(e squared x plus 1) minus 2arctge to the power of x
y equally ln(e squared x plus one) minus 2arctge to the power of x
y=ln(e2x+1)-2arctgex
y=lne2x+1-2arctgex
y=ln(e²x+1)-2arctge^x
y=ln(e to the power of 2x+1)-2arctge to the power of x
y=lne^2x+1-2arctge^x
Similar expressions
y=ln(e^2x+1)+2arctge^x
y=ln(e^2x-1)-2arctge^x
What you mean?
log(e^2*x + 1) - 2*atan(e)^x
log(e^(2*x) + 1) - 2*atan(e)^x
log(e^(2*x + 1)) - 2*atan(e)^x
log(e^(2*x + 1)) - 2*atan(e)^x

The derivative of the function/ y=ln(e^2x+1)-2arctge^x

You entered:

y=ln(e^2x+1)-2arctge^x

What you mean?

log(e^2*x + 1) - 2*atan(e)^x

log(e^(2*x) + 1) - 2*atan(e)^x

log(e^(2*x + 1)) - 2*atan(e)^x

log(e^(2*x + 1)) - 2*atan(e)^x

Derivative of y=ln(e^2x+1)-2arctge^x

The solution

You have entered [src]

   / 2      \         x   
log\e *x + 1/ - 2*atan (e)

$$\log{\left(x e^{2} + 1 \right)} - 2 \operatorname{atan}^{x}{\left(e \right)}$$

d /   / 2      \         x   \
--\log\e *x + 1/ - 2*atan (e)/
dx

$$\frac{d}{d x} \left(\log{\left(x e^{2} + 1 \right)} - 2 \operatorname{atan}^{x}{\left(e \right)}\right)$$

Transform

Detail solution

Differentiate $\log{\left(x e^{2} + 1 \right)} - 2 \operatorname{atan}^{x}{\left(e \right)}$ term by term:
1. Let $u = x e^{2} + 1$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x e^{2} + 1\right)$ :
  1. Differentiate $x e^{2} + 1$ term by term:
    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: $e^{2}$
    2. The derivative of the constant $1$ is zero.
    The result is: $e^{2}$
  The result of the chain rule is:
  
  $\frac{e^{2}}{x e^{2} + 1}$
4. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. $\frac{d}{d x} \operatorname{atan}^{x}{\left(e \right)} = \log{\left(\operatorname{atan}{\left(e \right)} \right)} \operatorname{atan}^{x}{\left(e \right)}$
    So, the result is: $2 \log{\left(\operatorname{atan}{\left(e \right)} \right)} \operatorname{atan}^{x}{\left(e \right)}$
  So, the result is: $- 2 \log{\left(\operatorname{atan}{\left(e \right)} \right)} \operatorname{atan}^{x}{\left(e \right)}$
The result is: $- 2 \log{\left(\operatorname{atan}{\left(e \right)} \right)} \operatorname{atan}^{x}{\left(e \right)} + \frac{e^{2}}{x e^{2} + 1}$
Now simplify:

$\frac{- 2 \left(x e^{2} + 1\right) \log{\left(\operatorname{atan}{\left(e \right)} \right)} \operatorname{atan}^{x}{\left(e \right)} + e^{2}}{x e^{2} + 1}$

The answer is:

$\frac{- 2 \left(x e^{2} + 1\right) \log{\left(\operatorname{atan}{\left(e \right)} \right)} \operatorname{atan}^{x}{\left(e \right)} + e^{2}}{x e^{2} + 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    2                             
   e             x                
-------- - 2*atan (e)*log(atan(e))
 2                                
e *x + 1

$$- 2 \log{\left(\operatorname{atan}{\left(e \right)} \right)} \operatorname{atan}^{x}{\left(e \right)} + \frac{e^{2}}{x e^{2} + 1}$$

Simplify

The second derivative [src]

 /      4                               \
 |     e              x       2         |
-|----------- + 2*atan (e)*log (atan(e))|
 |          2                           |
 |/       2\                            |
 \\1 + x*e /                            /

$$- (2 \log{\left(\operatorname{atan}{\left(e \right)} \right)}^{2} \operatorname{atan}^{x}{\left(e \right)} + \frac{e^{4}}{\left(x e^{2} + 1\right)^{2}})$$

Simplify

The third derivative [src]

  /      6                             \
  |     e            x       3         |
2*|----------- - atan (e)*log (atan(e))|
  |          3                         |
  |/       2\                          |
  \\1 + x*e /                          /

$$2 \left(- \log{\left(\operatorname{atan}{\left(e \right)} \right)}^{3} \operatorname{atan}^{x}{\left(e \right)} + \frac{e^{6}}{\left(x e^{2} + 1\right)^{3}}\right)$$

Simplify

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What you mean?

Derivative of y=ln(e^2x+1)-2arctge^x

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The solution