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Derivative of:
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Derivative of 1/tan(x)
Graphing y =:
log(x^2+x,2)
Identical expressions
log(x^ two +x, two)
logarithm of (x squared plus x,2)
logarithm of (x to the power of two plus x, two)
log(x2+x,2)
logx2+x,2
log(x²+x,2)
log(x to the power of 2+x,2)
logx^2+x,2
Similar expressions
log(x^2-x,2)

The derivative of the function/ log(x^2+x,2)

Derivative of log(x^2+x,2)

The solution

You have entered [src]

   / 2    \
log\x  + x/
-----------
   log(2)

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

log(x^2 + x)/log(2)

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of log(x^2+x,2)

The graph:

Piecewise:

The solution