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log3(3x)/3^2x-1

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log3(3x)/3^2x-1

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Derivative of log3(3x)/3^2x-1

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

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The solution

You have entered [src]
/log(3*x)\      
|--------|      
\ log(3) /      
----------*x - 1
    9           
x1log(3)log(3x)91x \frac{\frac{1}{\log{\left(3 \right)}} \log{\left(3 x \right)}}{9} - 1
Detail solution
  1. Differentiate x1log(3)log(3x)91x \frac{\frac{1}{\log{\left(3 \right)}} \log{\left(3 x \right)}}{9} - 1 term by term:

    1. Apply the quotient rule, which is:

      ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)g2(x)\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}

      f(x)=xlog(3x)f{\left(x \right)} = x \log{\left(3 x \right)} and g(x)=9log(3)g{\left(x \right)} = 9 \log{\left(3 \right)}.

      To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Apply the product rule:

        ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}

        f(x)=xf{\left(x \right)} = x; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

        1. Apply the power rule: xx goes to 11

        g(x)=log(3x)g{\left(x \right)} = \log{\left(3 x \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

        1. Let u=3xu = 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 ddx3x\frac{d}{d x} 3 x:

          1. 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 of the chain rule is:

          1x\frac{1}{x}

        The result is: log(3x)+1\log{\left(3 x \right)} + 1

      To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. The derivative of the constant 9log(3)9 \log{\left(3 \right)} is zero.

      Now plug in to the quotient rule:

      log(3x)+19log(3)\frac{\log{\left(3 x \right)} + 1}{9 \log{\left(3 \right)}}

    2. The derivative of the constant 1-1 is zero.

    The result is: log(3x)+19log(3)\frac{\log{\left(3 x \right)} + 1}{9 \log{\left(3 \right)}}


The answer is:

log(3x)+19log(3)\frac{\log{\left(3 x \right)} + 1}{9 \log{\left(3 \right)}}

The graph
02468-8-6-4-2-10105-5
The first derivative [src]
           /log(3*x)\
           |--------|
   1       \ log(3) /
-------- + ----------
9*log(3)       9     
1log(3)log(3x)9+19log(3)\frac{\frac{1}{\log{\left(3 \right)}} \log{\left(3 x \right)}}{9} + \frac{1}{9 \log{\left(3 \right)}}
The second derivative [src]
    1     
----------
9*x*log(3)
19xlog(3)\frac{1}{9 x \log{\left(3 \right)}}
The third derivative [src]
    -1     
-----------
   2       
9*x *log(3)
19x2log(3)- \frac{1}{9 x^{2} \log{\left(3 \right)}}
The graph
Derivative of log3(3x)/3^2x-1