Derivative of y=log8(x²+3x)

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   / 2      \
log\x  + 3*x/
-------------
    log(8)

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

log(x^2 + 3*x)/log(8)

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = x^{2} + 3 x$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} + 3 x\right)$ :
  1. Differentiate $x^{2} + 3 x$ term by term:
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $3$
    The result is: $2 x + 3$
  The result of the chain rule is:
  
  $\frac{2 x + 3}{x^{2} + 3 x}$
So, the result is: $\frac{2 x + 3}{\left(x^{2} + 3 x\right) \log{\left(8 \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     3 + 2*x     
-----------------
/ 2      \       
\x  + 3*x/*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)

$$\frac{2 - \frac{\left(2 x + 3\right)^{2}}{x \left(x + 3\right)}}{x \left(x + 3\right) \log{\left(8 \right)}}$$

Simplify

The third derivative [src]

             /             2\
             |    (3 + 2*x) |
-2*(3 + 2*x)*|3 - ----------|
             \    x*(3 + x) /
-----------------------------
       2        2            
      x *(3 + x) *log(8)

$$- \frac{2 \left(3 - \frac{\left(2 x + 3\right)^{2}}{x \left(x + 3\right)}\right) \left(2 x + 3\right)}{x^{2} \left(x + 3\right)^{2} \log{\left(8 \right)}}$$

Simplify

The graph

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Derivative of y=log8(x²+3x)

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