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Derivative of:
Derivative of x/(e^x+1)
Derivative of (x+1)^1/3
Derivative of sin(4*x-1)^(3)
Derivative of sin(9x+2)
Identical expressions
log(six ^sqrt(x))
logarithm of (6 to the power of square root of (x))
logarithm of (six to the power of square root of (x))
log(6^√(x))
log(6sqrt(x))
log6sqrtx
log6^sqrtx

The derivative of the function/ log(6^sqrt(x))

Derivative of log(6^sqrt(x))

The solution

You have entered [src]

   /   ___\
   | \/ x |
log\6     /

$$\log{\left(6^{\sqrt{x}} \right)}$$

log(6^(sqrt(x)))

Detail solution

Let $u = 6^{\sqrt{x}}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} 6^{\sqrt{x}}$ :
1. Let $u = \sqrt{x}$ .
2. $\frac{d}{d u} 6^{u} = 6^{u} \log{\left(6 \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{x}$ :
  1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
  The result of the chain rule is:
  
  $\frac{6^{\sqrt{x}} \log{\left(6 \right)}}{2 \sqrt{x}}$
The result of the chain rule is:

$\frac{\log{\left(6 \right)}}{2 \sqrt{x}}$

The answer is:

$\frac{\log{\left(6 \right)}}{2 \sqrt{x}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 log(6)
-------
    ___
2*\/ x

$$\frac{\log{\left(6 \right)}}{2 \sqrt{x}}$$

Simplify

The second derivative [src]

-log(6) 
--------
    3/2 
 4*x

$$- \frac{\log{\left(6 \right)}}{4 x^{\frac{3}{2}}}$$

Simplify

The third derivative [src]

3*log(6)
--------
    5/2 
 8*x

$$\frac{3 \log{\left(6 \right)}}{8 x^{\frac{5}{2}}}$$

Simplify