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ln(x+3)^3

Derivative of ln(x+3)^3

Function f() - derivative -N order at the point
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The solution

You have entered [src]
   3       
log (x + 3)
log(x+3)3\log{\left(x + 3 \right)}^{3}
d /   3       \
--\log (x + 3)/
dx             
ddxlog(x+3)3\frac{d}{d x} \log{\left(x + 3 \right)}^{3}
Detail solution
  1. Let u=log(x+3)u = \log{\left(x + 3 \right)}.

  2. Apply the power rule: u3u^{3} goes to 3u23 u^{2}

  3. Then, apply the chain rule. Multiply by ddxlog(x+3)\frac{d}{d x} \log{\left(x + 3 \right)}:

    1. Let u=x+3u = x + 3.

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

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

      1. Differentiate x+3x + 3 term by term:

        1. Apply the power rule: xx goes to 11

        2. The derivative of the constant 33 is zero.

        The result is: 11

      The result of the chain rule is:

      1x+3\frac{1}{x + 3}

    The result of the chain rule is:

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

  4. Now simplify:

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


The answer is:

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

The graph
02468-8-6-4-2-1010-500500
The first derivative [src]
     2       
3*log (x + 3)
-------------
    x + 3    
3log(x+3)2x+3\frac{3 \log{\left(x + 3 \right)}^{2}}{x + 3}
The second derivative [src]
3*(2 - log(3 + x))*log(3 + x)
-----------------------------
                  2          
           (3 + x)           
3(2log(x+3))log(x+3)(x+3)2\frac{3 \cdot \left(2 - \log{\left(x + 3 \right)}\right) \log{\left(x + 3 \right)}}{\left(x + 3\right)^{2}}
The third derivative [src]
  /       2                      \
6*\1 + log (3 + x) - 3*log(3 + x)/
----------------------------------
                    3             
             (3 + x)              
6(log(x+3)23log(x+3)+1)(x+3)3\frac{6 \left(\log{\left(x + 3 \right)}^{2} - 3 \log{\left(x + 3 \right)} + 1\right)}{\left(x + 3\right)^{3}}
The graph
Derivative of ln(x+3)^3