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Derivative of:
Derivative of (x^5+1)
Derivative of x^sqrt(x)
Derivative of log(x+2)
Derivative of ex^2
Identical expressions
y=ln^ five (sqrt(3x+ five))
y equally ln to the power of 5( square root of (3x plus 5))
y equally ln to the power of five ( square root of (3x plus five))
y=ln^5(√(3x+5))
y=ln5(sqrt(3x+5))
y=ln5sqrt3x+5
y=ln⁵(sqrt(3x+5))
y=ln^5sqrt3x+5
Similar expressions
y=ln^5(sqrt(3x-5))

The derivative of the function/ y=ln^5(sqrt(3x+5))

Derivative of y=ln^5(sqrt(3x+5))

The solution

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

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

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

Detail solution

Let $u = \log{\left(\sqrt{3 x + 5} \right)}$ .
Apply the power rule: $u^{5}$ goes to $5 u^{4}$
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:

$\frac{15 \log{\left(\sqrt{3 x + 5} \right)}^{4}}{2 \left(3 x + 5\right)}$
Now simplify:

$\frac{15 \log{\left(3 x + 5 \right)}^{4}}{32 \left(3 x + 5\right)}$

The answer is:

$\frac{15 \log{\left(3 x + 5 \right)}^{4}}{32 \left(3 x + 5\right)}$

The graph

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The first derivative [src]

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

$$\frac{15 \log{\left(\sqrt{3 x + 5} \right)}^{4}}{2 \left(3 x + 5\right)}$$

Simplify

The second derivative [src]

                     /       /  _________\\
      3/  _________\ |    log\\/ 5 + 3*x /|
45*log \\/ 5 + 3*x /*|1 - ----------------|
                     \           2        /
-------------------------------------------
                          2                
                 (5 + 3*x)

$$\frac{45 \left(1 - \frac{\log{\left(\sqrt{3 x + 5} \right)}}{2}\right) \log{\left(\sqrt{3 x + 5} \right)}^{3}}{\left(3 x + 5\right)^{2}}$$

Simplify

The third derivative [src]

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

$$\frac{135 \left(\log{\left(\sqrt{3 x + 5} \right)}^{2} - 3 \log{\left(\sqrt{3 x + 5} \right)} + \frac{3}{2}\right) \log{\left(\sqrt{3 x + 5} \right)}^{2}}{\left(3 x + 5\right)^{3}}$$

Simplify

The graph

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Derivative of y=ln^5(sqrt(3x+5))

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The solution