Derivative of y=ln(x²+5x-1)

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   / 2          \
log\x  + 5*x - 1/

$$\log{\left(\left(x^{2} + 5 x\right) - 1 \right)}$$

log(x^2 + 5*x - 1)

Detail solution

Let $u = \left(x^{2} + 5 x\right) - 1$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\left(x^{2} + 5 x\right) - 1\right)$ :
1. Differentiate $\left(x^{2} + 5 x\right) - 1$ term by term:
  1. Differentiate $x^{2} + 5 x$ term by term:
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    2. 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: $5$
    The result is: $2 x + 5$
  2. The derivative of the constant $-1$ is zero.
  The result is: $2 x + 5$
The result of the chain rule is:

$\frac{2 x + 5}{\left(x^{2} + 5 x\right) - 1}$
Now simplify:

$\frac{2 x + 5}{x^{2} + 5 x - 1}$

The answer is:

$\frac{2 x + 5}{x^{2} + 5 x - 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  5 + 2*x   
------------
 2          
x  + 5*x - 1

$$\frac{2 x + 5}{\left(x^{2} + 5 x\right) - 1}$$

Simplify

The second derivative [src]

               2 
      (5 + 2*x)  
2 - -------------
          2      
    -1 + x  + 5*x
-----------------
        2        
  -1 + x  + 5*x

$$\frac{- \frac{\left(2 x + 5\right)^{2}}{x^{2} + 5 x - 1} + 2}{x^{2} + 5 x - 1}$$

Simplify

The third derivative [src]

  /                2 \          
  |       (5 + 2*x)  |          
2*|-3 + -------------|*(5 + 2*x)
  |           2      |          
  \     -1 + x  + 5*x/          
--------------------------------
                       2        
        /      2      \         
        \-1 + x  + 5*x/

$$\frac{2 \left(2 x + 5\right) \left(\frac{\left(2 x + 5\right)^{2}}{x^{2} + 5 x - 1} - 3\right)}{\left(x^{2} + 5 x - 1\right)^{2}}$$

Simplify

The graph

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Derivative of y=ln(x²+5x-1)

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The solution