Derivative of 8^x^2

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 / 2\
 \x /
8

$$8^{x^{2}}$$

  / / 2\\
d | \x /|
--\8    /
dx

$$\frac{d}{d x} 8^{x^{2}}$$

Transform

Detail solution

Let $u = x^{2}$ .
$\frac{d}{d u} 8^{u} = 8^{u} \log{\left(8 \right)}$
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:

$2 \cdot 8^{x^{2}} x \log{\left(8 \right)}$
Now simplify:

$6 \cdot 2^{3 x^{2}} x \log{\left(2 \right)}$

The answer is:

$6 \cdot 2^{3 x^{2}} x \log{\left(2 \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     / 2\       
     \x /       
2*x*8    *log(8)

$$2 \cdot 8^{x^{2}} x \log{\left(8 \right)}$$

Simplify

The second derivative [src]

   / 2\                         
   \x / /       2       \       
2*8    *\1 + 2*x *log(8)/*log(8)

$$2 \cdot 8^{x^{2}} \cdot \left(2 x^{2} \log{\left(8 \right)} + 1\right) \log{\left(8 \right)}$$

Simplify

The third derivative [src]

     / 2\                          
     \x /    2    /       2       \
4*x*8    *log (8)*\3 + 2*x *log(8)/

$$4 \cdot 8^{x^{2}} x \left(2 x^{2} \log{\left(8 \right)} + 3\right) \log{\left(8 \right)}^{2}$$

Simplify

The graph