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y=x*sqrt(x)*(4ln(5x)-6)

Derivative of y=x*sqrt(x)*(4ln(5x)-6)

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

You have entered [src]
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x*\/ x *(4*log(5*x) - 6)
xx(4log(5x)6)\sqrt{x} x \left(4 \log{\left(5 x \right)} - 6\right)
(x*sqrt(x))*(4*log(5*x) - 6)
Detail solution
  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)=xxf{\left(x \right)} = \sqrt{x} x; 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)=xg{\left(x \right)} = \sqrt{x}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. Apply the power rule: x\sqrt{x} goes to 12x\frac{1}{2 \sqrt{x}}

      The result is: 3x2\frac{3 \sqrt{x}}{2}

    g(x)=4log(5x)6g{\left(x \right)} = 4 \log{\left(5 x \right)} - 6; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Differentiate 4log(5x)64 \log{\left(5 x \right)} - 6 term by term:

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

        1. Let u=5xu = 5 x.

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

        3. Then, apply the chain rule. Multiply by ddx5x\frac{d}{d x} 5 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: 55

          The result of the chain rule is:

          1x\frac{1}{x}

        So, the result is: 4x\frac{4}{x}

      2. The derivative of the constant 6-6 is zero.

      The result is: 4x\frac{4}{x}

    The result is: 3x(4log(5x)6)2+4x\frac{3 \sqrt{x} \left(4 \log{\left(5 x \right)} - 6\right)}{2} + 4 \sqrt{x}

  2. Now simplify:

    x(6log(5x)5)\sqrt{x} \left(6 \log{\left(5 x \right)} - 5\right)


The answer is:

x(6log(5x)5)\sqrt{x} \left(6 \log{\left(5 x \right)} - 5\right)

The graph
02468-8-6-4-2-1010-500500
The first derivative [src]
              ___                 
    ___   3*\/ x *(4*log(5*x) - 6)
4*\/ x  + ------------------------
                     2            
3x(4log(5x)6)2+4x\frac{3 \sqrt{x} \left(4 \log{\left(5 x \right)} - 6\right)}{2} + 4 \sqrt{x}
The second derivative [src]
7/2 + 3*log(5*x)
----------------
       ___      
     \/ x       
3log(5x)+72x\frac{3 \log{\left(5 x \right)} + \frac{7}{2}}{\sqrt{x}}
The third derivative [src]
-(-5 + 6*log(5*x)) 
-------------------
          3/2      
       4*x         
6log(5x)54x32- \frac{6 \log{\left(5 x \right)} - 5}{4 x^{\frac{3}{2}}}
The graph
Derivative of y=x*sqrt(x)*(4ln(5x)-6)