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y=log5(x)-√xx^6
The derivative of the function/ y=log5(x)-√xx^6

Derivative of y=log5(x)-√xx^6

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

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The solution

You have entered [src] 
                6
log(x)     _____ 
------ - \/ x*x  
log(5)
−(xx)6+log⁡(x)log⁡(5)- \left(\sqrt{x x}\right)^{6} + \frac{\log{\left(x \right)}}{\log{\left(5 \right)}} 
  /                6\
d |log(x)     _____ |
--|------ - \/ x*x  |
dx\log(5)           /
ddx(−(xx)6+log⁡(x)log⁡(5))\frac{d}{d x} \left(- \left(\sqrt{x x}\right)^{6} + \frac{\log{\left(x \right)}}{\log{\left(5 \right)}}\right)
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Detail solution

  1. Differentiate −(xx)6+log⁡(x)log⁡(5)- \left(\sqrt{x x}\right)^{6} + \frac{\log{\left(x \right)}}{\log{\left(5 \right)}} term by term:

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

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

      So, the result is: 1xlog⁡(5)\frac{1}{x \log{\left(5 \right)}}

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

      1. Let u=xxu = \sqrt{x x}.

      2. Apply the power rule: u6u^{6} goes to 6u56 u^{5}

      3. Then, apply the chain rule. Multiply by ddxxx\frac{d}{d x} \sqrt{x x}:

        1. Let u=xxu = x x.

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

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

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

            1. Apply the power rule: xx goes to 11

            The result is: 2x2 x

          The result of the chain rule is:

          x∣x∣\frac{x}{\left|{x}\right|}

        The result of the chain rule is:

        6x56 x^{5}

      So, the result is: −6x5- 6 x^{5}

    The result is: −6x5+1xlog⁡(5)- 6 x^{5} + \frac{1}{x \log{\left(5 \right)}}

The answer is:

−6x5+1xlog⁡(5)- 6 x^{5} + \frac{1}{x \log{\left(5 \right)}}

The graph
02468-8-6-4-2-1010-20002000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
     5      1    
- 6*x  + --------
         x*log(5)
−6x5+1xlog⁡(5)- 6 x^{5} + \frac{1}{x \log{\left(5 \right)}}
Simplify
The second derivative [src] 
 /    4       1    \
-|30*x  + ---------|
 |         2       |
 \        x *log(5)/
−(30x4+1x2log⁡(5))- (30 x^{4} + \frac{1}{x^{2} \log{\left(5 \right)}})
Simplify
The third derivative [src] 
  /      3       1    \
2*|- 60*x  + ---------|
  |           3       |
  \          x *log(5)/
2(−60x3+1x3log⁡(5))2 \left(- 60 x^{3} + \frac{1}{x^{3} \log{\left(5 \right)}}\right)
Simplify
The graph
Derivative of y=log5(x)-√xx^6
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