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y=log8(x^2+3x)
The derivative of the function/ y=log8(x^2+3x)

Derivative of y=log8(x^2+3x)

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

You have entered [src] 
   / 2      \
log\x  + 3*x/
-------------
    log(8)
log⁡(x2+3x)log⁡(8)\frac{\log{\left(x^{2} + 3 x \right)}}{\log{\left(8 \right)}} 
  /   / 2      \\
d |log\x  + 3*x/|
--|-------------|
dx\    log(8)   /
ddxlog⁡(x2+3x)log⁡(8)\frac{d}{d x} \frac{\log{\left(x^{2} + 3 x \right)}}{\log{\left(8 \right)}}
Transform
Detail solution

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

    1. Let u=x2+3xu = x^{2} + 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(x2+3x)\frac{d}{d x} \left(x^{2} + 3 x\right):

      1. Differentiate x2+3xx^{2} + 3 x term by term:

        1. Apply the power rule: x2x^{2} goes to 2x2 x

        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: 2x+32 x + 3

      The result of the chain rule is:

      2x+3x2+3x\frac{2 x + 3}{x^{2} + 3 x}

    So, the result is: 2x+3(x2+3x)log⁡(8)\frac{2 x + 3}{\left(x^{2} + 3 x\right) \log{\left(8 \right)}}

  2. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-1010
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
     3 + 2*x     
-----------------
/ 2      \       
\x  + 3*x/*log(8)
2x+3(x2+3x)log⁡(8)\frac{2 x + 3}{\left(x^{2} + 3 x\right) \log{\left(8 \right)}}
Simplify
The second derivative [src] 
              2 
     (3 + 2*x)  
 2 - ---------- 
     x*(3 + x)  
----------------
x*(3 + x)*log(8)
2−(2x+3)2x(x+3)x(x+3)log⁡(8)\frac{2 - \frac{\left(2 x + 3\right)^{2}}{x \left(x + 3\right)}}{x \left(x + 3\right) \log{\left(8 \right)}}
Simplify
The third derivative [src] 
             /             2\
             |    (3 + 2*x) |
-2*(3 + 2*x)*|3 - ----------|
             \    x*(3 + x) /
-----------------------------
       2        2            
      x *(3 + x) *log(8)
−2⋅(3−(2x+3)2x(x+3))(2x+3)x2(x+3)2log⁡(8)- \frac{2 \cdot \left(3 - \frac{\left(2 x + 3\right)^{2}}{x \left(x + 3\right)}\right) \left(2 x + 3\right)}{x^{2} \left(x + 3\right)^{2} \log{\left(8 \right)}}
Simplify
The graph
Derivative of y=log8(x^2+3x)
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