Derivative of 5ln(3x+5ln(x))

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5*log(3*x + 5*log(x))

$$5 \log{\left(3 x + 5 \log{\left(x \right)} \right)}$$

5*log(3*x + 5*log(x))

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = 3 x + 5 \log{\left(x \right)}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(3 x + 5 \log{\left(x \right)}\right)$ :
  1. Differentiate $3 x + 5 \log{\left(x \right)}$ term by term:
    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: $3$
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
      So, the result is: $\frac{5}{x}$
    The result is: $3 + \frac{5}{x}$
  The result of the chain rule is:
  
  $\frac{3 + \frac{5}{x}}{3 x + 5 \log{\left(x \right)}}$
So, the result is: $\frac{5 \left(3 + \frac{5}{x}\right)}{3 x + 5 \log{\left(x \right)}}$
Now simplify:

$\frac{5 \left(3 x + 5\right)}{x \left(3 x + 5 \log{\left(x \right)}\right)}$

The answer is:

$\frac{5 \left(3 x + 5\right)}{x \left(3 x + 5 \log{\left(x \right)}\right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    /    5\   
  5*|3 + -|   
    \    x/   
--------------
3*x + 5*log(x)

$$\frac{5 \left(3 + \frac{5}{x}\right)}{3 x + 5 \log{\left(x \right)}}$$

Simplify

The second derivative [src]

   /               2   \
   |        /    5\    |
   |        |3 + -|    |
   |5       \    x/    |
-5*|-- + --------------|
   | 2   3*x + 5*log(x)|
   \x                  /
------------------------
     3*x + 5*log(x)

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

Simplify

The third derivative [src]

  /                  3                         \
  |           /    5\               /    5\    |
  |         2*|3 + -|            15*|3 + -|    |
  |10         \    x/               \    x/    |
5*|-- + ----------------- + -------------------|
  | 3                   2    2                 |
  \x    (3*x + 5*log(x))    x *(3*x + 5*log(x))/
------------------------------------------------
                 3*x + 5*log(x)

$$\frac{5 \left(\frac{2 \left(3 + \frac{5}{x}\right)^{3}}{\left(3 x + 5 \log{\left(x \right)}\right)^{2}} + \frac{15 \left(3 + \frac{5}{x}\right)}{x^{2} \left(3 x + 5 \log{\left(x \right)}\right)} + \frac{10}{x^{3}}\right)}{3 x + 5 \log{\left(x \right)}}$$

Simplify

The graph

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Derivative of 5ln(3x+5ln(x))

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Piecewise:

The solution