Derivative of 2^x^2

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 / 2\
 \x /
2

2^{x^{2}}

  / / 2\\
d | \x /|
--\2    /
dx

\frac{d}{d x} 2^{x^{2}}

Transform

Detail solution

Let $u = x^{2}$ .
$\frac{d}{d u} 2^{u} = 2^{u} \log{\left(2 \right)}$
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:

$2 \cdot 2^{x^{2}} x \log{\left(2 \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     / 2\       
     \x /       
2*x*2    *log(2)

2 \cdot 2^{x^{2}} x \log{\left(2 \right)}

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

     / 2\                          
     \x /    2    /       2       \
4*x*2    *log (2)*\3 + 2*x *log(2)/

4 \cdot 2^{x^{2}} x \left(2 x^{2} \log{\left(2 \right)} + 3\right) \log{\left(2 \right)}^{2}

Simplify

The graph