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Derivative of sin(4x)
Derivative of x^(sin(x)^(2))
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Derivative of log(x+(1+x^2)^(1/2))
Identical expressions
log6^sin4x*sqrt(x)
logarithm of 6 to the power of sinus of 4x multiply by square root of (x)
log6^sin4x*√(x)
log6sin4x*sqrt(x)
log6sin4x*sqrtx
log6^sin4xsqrt(x)
log6sin4xsqrt(x)
log6sin4xsqrtx
log6^sin4xsqrtx

The derivative of the function/ log6^sin4x*sqrt(x)

Derivative of log6^sin4x*sqrt(x)

The solution

You have entered [src]

   sin(4*x)      ___
log        (6)*\/ x

$$\sqrt{x} \log{\left(6 \right)}^{\sin{\left(4 x \right)}}$$

log(6)^sin(4*x)*sqrt(x)

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = \log{\left(6 \right)}^{\sin{\left(4 x \right)}}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \sin{\left(4 x \right)}$ .
2. $\frac{d}{d u} \log{\left(6 \right)}^{u} = \log{\left(6 \right)}^{u} \log{\left(\log{\left(6 \right)} \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(4 x \right)}$ :
  1. Let $u = 4 x$ .
  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 $\frac{d}{d x} 4 x$ :
    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: $4$
    The result of the chain rule is:
    
    $4 \cos{\left(4 x \right)}$
  The result of the chain rule is:
  
  $4 \log{\left(6 \right)}^{\sin{\left(4 x \right)}} \log{\left(\log{\left(6 \right)} \right)} \cos{\left(4 x \right)}$
$g{\left(x \right)} = \sqrt{x}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
The result is: $4 \sqrt{x} \log{\left(6 \right)}^{\sin{\left(4 x \right)}} \log{\left(\log{\left(6 \right)} \right)} \cos{\left(4 x \right)} + \frac{\log{\left(6 \right)}^{\sin{\left(4 x \right)}}}{2 \sqrt{x}}$
Now simplify:

$\frac{\left(8 x \log{\left(\log{\left(6 \right)} \right)} \cos{\left(4 x \right)} + 1\right) \log{\left(6 \right)}^{\sin{\left(4 x \right)}}}{2 \sqrt{x}}$

The answer is:

$\frac{\left(8 x \log{\left(\log{\left(6 \right)} \right)} \cos{\left(4 x \right)} + 1\right) \log{\left(6 \right)}^{\sin{\left(4 x \right)}}}{2 \sqrt{x}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   sin(4*x)                                                 
log        (6)       ___    sin(4*x)                        
-------------- + 4*\/ x *log        (6)*cos(4*x)*log(log(6))
       ___                                                  
   2*\/ x

$$4 \sqrt{x} \log{\left(6 \right)}^{\sin{\left(4 x \right)}} \log{\left(\log{\left(6 \right)} \right)} \cos{\left(4 x \right)} + \frac{\log{\left(6 \right)}^{\sin{\left(4 x \right)}}}{2 \sqrt{x}}$$

Simplify

The second derivative [src]

   sin(4*x)    /    1           ___ /     2                            \               4*cos(4*x)*log(log(6))\
log        (6)*|- ------ - 16*\/ x *\- cos (4*x)*log(log(6)) + sin(4*x)/*log(log(6)) + ----------------------|
               |     3/2                                                                         ___         |
               \  4*x                                                                          \/ x          /

$$\left(- 16 \sqrt{x} \left(\sin{\left(4 x \right)} - \log{\left(\log{\left(6 \right)} \right)} \cos^{2}{\left(4 x \right)}\right) \log{\left(\log{\left(6 \right)} \right)} + \frac{4 \log{\left(\log{\left(6 \right)} \right)} \cos{\left(4 x \right)}}{\sqrt{x}} - \frac{1}{4 x^{\frac{3}{2}}}\right) \log{\left(6 \right)}^{\sin{\left(4 x \right)}}$$

Simplify

The third derivative [src]

               /            /     2                            \                                                                                                                           \
   sin(4*x)    |  3      24*\- cos (4*x)*log(log(6)) + sin(4*x)/*log(log(6))   3*cos(4*x)*log(log(6))        ___ /       2         2                                 \                     |
log        (6)*|------ - --------------------------------------------------- - ---------------------- - 64*\/ x *\1 - cos (4*x)*log (log(6)) + 3*log(log(6))*sin(4*x)/*cos(4*x)*log(log(6))|
               |   5/2                            ___                                    3/2                                                                                               |
               \8*x                             \/ x                                    x                                                                                                  /

$$\left(- 64 \sqrt{x} \left(3 \log{\left(\log{\left(6 \right)} \right)} \sin{\left(4 x \right)} - \log{\left(\log{\left(6 \right)} \right)}^{2} \cos^{2}{\left(4 x \right)} + 1\right) \log{\left(\log{\left(6 \right)} \right)} \cos{\left(4 x \right)} - \frac{24 \left(\sin{\left(4 x \right)} - \log{\left(\log{\left(6 \right)} \right)} \cos^{2}{\left(4 x \right)}\right) \log{\left(\log{\left(6 \right)} \right)}}{\sqrt{x}} - \frac{3 \log{\left(\log{\left(6 \right)} \right)} \cos{\left(4 x \right)}}{x^{\frac{3}{2}}} + \frac{3}{8 x^{\frac{5}{2}}}\right) \log{\left(6 \right)}^{\sin{\left(4 x \right)}}$$

Simplify

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Derivative of log6^sin4x*sqrt(x)

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The solution