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Identical expressions
y=x^ one / two *ln3x
y equally x to the power of 1 divide by 2 multiply by ln3x
y equally x to the power of one divide by two multiply by ln3x
y=x1/2*ln3x
y=x^1/2ln3x
y=x1/2ln3x
y=x^1 divide by 2*ln3x

The derivative of the function/ y=x^1/2*ln3x

Derivative of y=x^1/2*ln3x

The solution

You have entered [src]

  ___         
\/ x *log(3*x)

$$\sqrt{x} \log{\left(3 x \right)}$$

d /  ___         \
--\\/ x *log(3*x)/
dx

$$\frac{d}{d x} \sqrt{x} \log{\left(3 x \right)}$$

Transform

Detail solution

Apply the product rule:

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

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

$\frac{\log{\left(3 x \right)} + 2}{2 \sqrt{x}}$

The answer is:

$\frac{\log{\left(3 x \right)} + 2}{2 \sqrt{x}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  1     log(3*x)
----- + --------
  ___       ___ 
\/ x    2*\/ x

$$\frac{\log{\left(3 x \right)}}{2 \sqrt{x}} + \frac{1}{\sqrt{x}}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Derivative of y=x^1/2*ln3x