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Derivative of:
Derivative of e^2*x
Derivative of cos(x)^2
Derivative of x^sin(x)
Derivative of sin(x^3)
Identical expressions
log(four *x+ five , three)
logarithm of (4 multiply by x plus 5,3)
logarithm of (four multiply by x plus five , three)
log(4x+5,3)
log4x+5,3
Similar expressions
log(4*x-5,3)

The derivative of the function/ log(4*x+5,3)

Derivative of log(4*x+5,3)

The solution

You have entered [src]

   /      53\
log|4*x + --|
   \      10/

$$\log{\left(4 x + \frac{53}{10} \right)}$$

log(4*x + 53/10)

Detail solution

Let $u = 4 x + \frac{53}{10}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(4 x + \frac{53}{10}\right)$ :
1. Differentiate $4 x + \frac{53}{10}$ 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: $4$
  2. The derivative of the constant $\frac{53}{10}$ is zero.
  The result is: $4$
The result of the chain rule is:

$\frac{4}{4 x + \frac{53}{10}}$
Now simplify:

$\frac{40}{40 x + 53}$

The answer is:

$\frac{40}{40 x + 53}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   4    
--------
      53
4*x + --
      10

$$\frac{4}{4 x + \frac{53}{10}}$$

Simplify

The second derivative [src]

    -16    
-----------
          2
/53      \ 
|-- + 4*x| 
\10      /

$$- \frac{16}{\left(4 x + \frac{53}{10}\right)^{2}}$$

Simplify

The third derivative [src]

    128    
-----------
          3
/53      \ 
|-- + 4*x| 
\10      /

$$\frac{128}{\left(4 x + \frac{53}{10}\right)^{3}}$$

Simplify