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Identical expressions
y=log(3x+ one)^ ten
y equally logarithm of (3x plus 1) to the power of 10
y equally logarithm of (3x plus one) to the power of ten
y=log(3x+1)10
y=log3x+110
y=log3x+1^10
Similar expressions
y=log(3x-1)^10

The derivative of the function/ y=log(3x+1)^10

Derivative of y=log(3x+1)^10

The solution

You have entered [src]

   10         
log  (3*x + 1)

$$\log{\left(3 x + 1 \right)}^{10}$$

log(3*x + 1)^10

Detail solution

Let $u = \log{\left(3 x + 1 \right)}$ .
Apply the power rule: $u^{10}$ goes to $10 u^{9}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(3 x + 1 \right)}$ :
1. Let $u = 3 x + 1$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(3 x + 1\right)$ :
  1. Differentiate $3 x + 1$ term by term:
    1. 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: $3$
    2. The derivative of the constant $1$ is zero.
    The result is: $3$
  The result of the chain rule is:
  
  $\frac{3}{3 x + 1}$
The result of the chain rule is:

$\frac{30 \log{\left(3 x + 1 \right)}^{9}}{3 x + 1}$
Now simplify:

$\frac{30 \log{\left(3 x + 1 \right)}^{9}}{3 x + 1}$

The answer is:

$\frac{30 \log{\left(3 x + 1 \right)}^{9}}{3 x + 1}$

The graph

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Plot the graph f'(x)

The first derivative [src]

      9         
30*log (3*x + 1)
----------------
    3*x + 1

$$\frac{30 \log{\left(3 x + 1 \right)}^{9}}{3 x + 1}$$

Simplify

The second derivative [src]

      8                            
90*log (1 + 3*x)*(9 - log(1 + 3*x))
-----------------------------------
                      2            
             (1 + 3*x)

$$\frac{90 \left(9 - \log{\left(3 x + 1 \right)}\right) \log{\left(3 x + 1 \right)}^{8}}{\left(3 x + 1\right)^{2}}$$

Simplify

The third derivative [src]

       7          /                            2         \
270*log (1 + 3*x)*\72 - 27*log(1 + 3*x) + 2*log (1 + 3*x)/
----------------------------------------------------------
                                 3                        
                        (1 + 3*x)

$$\frac{270 \left(2 \log{\left(3 x + 1 \right)}^{2} - 27 \log{\left(3 x + 1 \right)} + 72\right) \log{\left(3 x + 1 \right)}^{7}}{\left(3 x + 1\right)^{3}}$$

Simplify

The graph

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Derivative of y=log(3x+1)^10

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The solution