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The derivative of the function/ log9(x+3x)^2

Derivative of log9(x+3x)^2

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

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The solution

You have entered [src] 
              2
/log(x + 3*x)\ 
|------------| 
\   log(9)   /
(log⁡(x+3x)log⁡(9))2\left(\frac{\log{\left(x + 3 x \right)}}{\log{\left(9 \right)}}\right)^{2} 
(log(x + 3*x)/log(9))^2
Detail solution

  1. Let u=log⁡(x+3x)log⁡(9)u = \frac{\log{\left(x + 3 x \right)}}{\log{\left(9 \right)}}.

  2. Apply the power rule: u2u^{2} goes to 2u2 u

  3. Then, apply the chain rule. Multiply by ddxlog⁡(x+3x)log⁡(9)\frac{d}{d x} \frac{\log{\left(x + 3 x \right)}}{\log{\left(9 \right)}}:

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

      1. Let u=x+3xu = x + 3 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(x+3x)\frac{d}{d x} \left(x + 3 x\right):

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

          1. Apply the power rule: xx goes to 11

          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: 33

          The result is: 44

        The result of the chain rule is:

        4x+3x\frac{4}{x + 3 x}

      So, the result is: 4(x+3x)log⁡(9)\frac{4}{\left(x + 3 x\right) \log{\left(9 \right)}}

    The result of the chain rule is:

    8log⁡(x+3x)(x+3x)log⁡(9)2\frac{8 \log{\left(x + 3 x \right)}}{\left(x + 3 x\right) \log{\left(9 \right)}^{2}}

  4. Now simplify:

    2log⁡(4x)xlog⁡(9)2\frac{2 \log{\left(4 x \right)}}{x \log{\left(9 \right)}^{2}}

The answer is:

2log⁡(4x)xlog⁡(9)2\frac{2 \log{\left(4 x \right)}}{x \log{\left(9 \right)}^{2}}

The graph
02468-8-6-4-2-10105-5
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
        2             
     log (x + 3*x)    
   8*-------------    
           2          
        log (9)       
----------------------
(x + 3*x)*log(x + 3*x)
8log⁡(x+3x)2log⁡(9)2(x+3x)log⁡(x+3x)\frac{8 \frac{\log{\left(x + 3 x \right)}^{2}}{\log{\left(9 \right)}^{2}}}{\left(x + 3 x\right) \log{\left(x + 3 x \right)}}
Simplify
The second derivative [src] 
2*(1 - log(4*x))
----------------
    2    2      
   x *log (9)
2(1−log⁡(4x))x2log⁡(9)2\frac{2 \left(1 - \log{\left(4 x \right)}\right)}{x^{2} \log{\left(9 \right)}^{2}}
Simplify
The third derivative [src] 
2*(-3 + 2*log(4*x))
-------------------
      3    2       
     x *log (9)
2(2log⁡(4x)−3)x3log⁡(9)2\frac{2 \left(2 \log{\left(4 x \right)} - 3\right)}{x^{3} \log{\left(9 \right)}^{2}}
Simplify
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