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Identical expressions
y=log^ eight (x^ two +3x)
y equally logarithm of to the power of 8(x squared plus 3x)
y equally logarithm of to the power of eight (x to the power of two plus 3x)
y=log8(x2+3x)
y=log8x2+3x
y=log⁸(x²+3x)
y=log to the power of 8(x to the power of 2+3x)
y=log^8x^2+3x
Similar expressions
y=log^8(x^2-3x)

The derivative of the function/ y=log^8(x^2+3x)

Derivative of y=log^8(x^2+3x)

The solution

You have entered [src]

   8/ 2      \
log \x  + 3*x/

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

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

Detail solution

Let $u = \log{\left(x^{2} + 3 x \right)}$ .
Apply the power rule: $u^{8}$ goes to $8 u^{7}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x^{2} + 3 x \right)}$ :
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:
  
  $\frac{2 x + 3}{x^{2} + 3 x}$
The result of the chain rule is:

$\frac{8 \left(2 x + 3\right) \log{\left(x^{2} + 3 x \right)}^{7}}{x^{2} + 3 x}$
Now simplify:

$\frac{\left(16 x + 24\right) \log{\left(x \left(x + 3\right) \right)}^{7}}{x \left(x + 3\right)}$

The answer is:

$\frac{\left(16 x + 24\right) \log{\left(x \left(x + 3\right) \right)}^{7}}{x \left(x + 3\right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     7/ 2      \          
8*log \x  + 3*x/*(3 + 2*x)
--------------------------
          2               
         x  + 3*x

$$\frac{8 \left(2 x + 3\right) \log{\left(x^{2} + 3 x \right)}^{7}}{x^{2} + 3 x}$$

Simplify

The second derivative [src]

                  /                              2            2               \
     6            |                   7*(3 + 2*x)    (3 + 2*x) *log(x*(3 + x))|
8*log (x*(3 + x))*|2*log(x*(3 + x)) + ------------ - -------------------------|
                  \                    x*(3 + x)             x*(3 + x)        /
-------------------------------------------------------------------------------
                                   x*(3 + x)

$$\frac{8 \left(2 \log{\left(x \left(x + 3\right) \right)} - \frac{\left(2 x + 3\right)^{2} \log{\left(x \left(x + 3\right) \right)}}{x \left(x + 3\right)} + \frac{7 \left(2 x + 3\right)^{2}}{x \left(x + 3\right)}\right) \log{\left(x \left(x + 3\right) \right)}^{6}}{x \left(x + 3\right)}$$

Simplify

The third derivative [src]

                            /                                                      2               2                             2    2           \
     5                      |       2                                  42*(3 + 2*x)    21*(3 + 2*x) *log(x*(3 + x))   2*(3 + 2*x) *log (x*(3 + x))|
8*log (x*(3 + x))*(3 + 2*x)*|- 6*log (x*(3 + x)) + 42*log(x*(3 + x)) + ------------- - ---------------------------- + ----------------------------|
                            \                                            x*(3 + x)              x*(3 + x)                      x*(3 + x)          /
---------------------------------------------------------------------------------------------------------------------------------------------------
                                                                     2        2                                                                    
                                                                    x *(3 + x)

$$\frac{8 \left(2 x + 3\right) \left(- 6 \log{\left(x \left(x + 3\right) \right)}^{2} + 42 \log{\left(x \left(x + 3\right) \right)} + \frac{2 \left(2 x + 3\right)^{2} \log{\left(x \left(x + 3\right) \right)}^{2}}{x \left(x + 3\right)} - \frac{21 \left(2 x + 3\right)^{2} \log{\left(x \left(x + 3\right) \right)}}{x \left(x + 3\right)} + \frac{42 \left(2 x + 3\right)^{2}}{x \left(x + 3\right)}\right) \log{\left(x \left(x + 3\right) \right)}^{5}}{x^{2} \left(x + 3\right)^{2}}$$

Simplify

The graph

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Derivative of y=log^8(x^2+3x)

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Piecewise:

The solution