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

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

Function f() - derivative -N order at the point
v

The graph:

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Piecewise:

The solution

You have entered [src] 
   / 2      \
log\x  + 2*x/
log⁡(x2+2x)\log{\left(x^{2} + 2 x \right)} 
d /   / 2      \\
--\log\x  + 2*x//
dx
ddxlog⁡(x2+2x)\frac{d}{d x} \log{\left(x^{2} + 2 x \right)}
Transform
Detail solution

  1. Let u=x2+2xu = x^{2} + 2 x.

  2. The derivative of log⁡(u)\log{\left(u \right)} is 1u\frac{1}{u}.

  3. Then, apply the chain rule. Multiply by ddx(x2+2x)\frac{d}{d x} \left(x^{2} + 2 x\right):

    1. Differentiate x2+2xx^{2} + 2 x term by term:

      1. Apply the power rule: x2x^{2} goes to 2x2 x

      2. The derivative of a constant times a function is the constant times the derivative of the function.

        1. Apply the power rule: xx goes to 11

        So, the result is: 22

      The result is: 2x+22 x + 2

    The result of the chain rule is:

    2x+2x2+2x\frac{2 x + 2}{x^{2} + 2 x}

  4. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-2525
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
2 + 2*x 
--------
 2      
x  + 2*x
2x+2x2+2x\frac{2 x + 2}{x^{2} + 2 x}
Simplify
The second derivative [src] 
  /             2\
  |    2*(1 + x) |
2*|1 - ----------|
  \    x*(2 + x) /
------------------
    x*(2 + x)
2⋅(1−2(x+1)2x(x+2))x(x+2)\frac{2 \cdot \left(1 - \frac{2 \left(x + 1\right)^{2}}{x \left(x + 2\right)}\right)}{x \left(x + 2\right)}
Simplify
The third derivative [src] 
          /              2\
          |     4*(1 + x) |
4*(1 + x)*|-3 + ----------|
          \     x*(2 + x) /
---------------------------
         2        2        
        x *(2 + x)
4(−3+4(x+1)2x(x+2))(x+1)x2(x+2)2\frac{4 \left(-3 + \frac{4 \left(x + 1\right)^{2}}{x \left(x + 2\right)}\right) \left(x + 1\right)}{x^{2} \left(x + 2\right)^{2}}
Simplify
The graph
Derivative of log(x^2+2*x)
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