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Identical expressions
ln^ four *(sqrt(three *x+ five))
ln to the power of 4 multiply by ( square root of (3 multiply by x plus 5))
ln to the power of four multiply by ( square root of (three multiply by x plus five))
ln^4*(√(3*x+5))
ln4*(sqrt(3*x+5))
ln4*sqrt3*x+5
ln⁴*(sqrt(3*x+5))
ln^4(sqrt(3x+5))
ln4(sqrt(3x+5))
ln4sqrt3x+5
ln^4sqrt3x+5
Similar expressions
ln^4*(sqrt(3*x-5))

The derivative of the function/ ln^4*(sqrt(3*x+5))

Derivative of ln^4(sqrt(3x+5))

The solution

You have entered [src]

   4/  _________\
log \\/ 3*x + 5 /

$$\log{\left(\sqrt{3 x + 5} \right)}^{4}$$

log(sqrt(3*x + 5))^4

Detail solution

Let $u = \log{\left(\sqrt{3 x + 5} \right)}$ .
Apply the power rule: $u^{4}$ goes to $4 u^{3}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(\sqrt{3 x + 5} \right)}$ :
1. Let $u = \sqrt{3 x + 5}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{3 x + 5}$ :
  1. Let $u = 3 x + 5$ .
  2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(3 x + 5\right)$ :
    1. Differentiate $3 x + 5$ term by term:
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $3$
      2. The derivative of the constant $5$ is zero.
      The result is: $3$
    The result of the chain rule is:
    
    $\frac{3}{2 \sqrt{3 x + 5}}$
  The result of the chain rule is:
  
  $\frac{3}{2 \left(3 x + 5\right)}$
The result of the chain rule is:

$\frac{6 \log{\left(\sqrt{3 x + 5} \right)}^{3}}{3 x + 5}$
Now simplify:

$\frac{3 \log{\left(3 x + 5 \right)}^{3}}{4 \left(3 x + 5\right)}$

The answer is:

$\frac{3 \log{\left(3 x + 5 \right)}^{3}}{4 \left(3 x + 5\right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     3/  _________\
6*log \\/ 3*x + 5 /
-------------------
      3*x + 5

$$\frac{6 \log{\left(\sqrt{3 x + 5} \right)}^{3}}{3 x + 5}$$

Simplify

The second derivative [src]

     2/  _________\ /         /  _________\\
9*log \\/ 5 + 3*x /*\3 - 2*log\\/ 5 + 3*x //
--------------------------------------------
                          2                 
                 (5 + 3*x)

$$\frac{9 \left(3 - 2 \log{\left(\sqrt{3 x + 5} \right)}\right) \log{\left(\sqrt{3 x + 5} \right)}^{2}}{\left(3 x + 5\right)^{2}}$$

Simplify

The third derivative [src]

   /         /  _________\        2/  _________\\    /  _________\
27*\3 - 9*log\\/ 5 + 3*x / + 4*log \\/ 5 + 3*x //*log\\/ 5 + 3*x /
------------------------------------------------------------------
                                     3                            
                            (5 + 3*x)

$$\frac{27 \left(4 \log{\left(\sqrt{3 x + 5} \right)}^{2} - 9 \log{\left(\sqrt{3 x + 5} \right)} + 3\right) \log{\left(\sqrt{3 x + 5} \right)}}{\left(3 x + 5\right)^{3}}$$

Simplify

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Derivative of ln^4(sqrt(3x+5))

The graph:

Piecewise:

The solution

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Derivative of ln^4*(sqrt(3*x+5))

The graph:

Piecewise:

The solution

Derivative of ln^4(sqrt(3x+5))