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Derivative of cbrt(6x^2+5x)
Identical expressions
cbrt(6x^ two +5x)
cubic root of (6x squared plus 5x)
cubic root of (6x to the power of two plus 5x)
cbrt(6x2+5x)
cbrt6x2+5x
cbrt(6x²+5x)
cbrt(6x to the power of 2+5x)
cbrt6x^2+5x
Similar expressions
cbrt(6x^2-5x)

The derivative of the function/ cbrt(6x^2+5x)

Derivative of cbrt(6x^2+5x)

The solution

You have entered [src]

   ____________
3 /    2       
\/  6*x  + 5*x

$$\sqrt[3]{6 x^{2} + 5 x}$$

(6*x^2 + 5*x)^(1/3)

Detail solution

Let $u = 6 x^{2} + 5 x$ .
Apply the power rule: $\sqrt[3]{u}$ goes to $\frac{1}{3 u^{\frac{2}{3}}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(6 x^{2} + 5 x\right)$ :
1. Differentiate $6 x^{2} + 5 x$ 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: $x^{2}$ goes to $2 x$
    So, the result is: $12 x$
  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: $5$
  The result is: $12 x + 5$
The result of the chain rule is:

$\frac{12 x + 5}{3 \left(6 x^{2} + 5 x\right)^{\frac{2}{3}}}$
Now simplify:

$\frac{12 x + 5}{3 \left(x \left(6 x + 5\right)\right)^{\frac{2}{3}}}$

The answer is:

$\frac{12 x + 5}{3 \left(x \left(6 x + 5\right)\right)^{\frac{2}{3}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   5/3 + 4*x   
---------------
            2/3
/   2      \   
\6*x  + 5*x/

$$\frac{4 x + \frac{5}{3}}{\left(6 x^{2} + 5 x\right)^{\frac{2}{3}}}$$

Simplify

The second derivative [src]

  /               2 \
  |     (5 + 12*x)  |
2*|2 - -------------|
  \    9*x*(5 + 6*x)/
---------------------
                2/3  
   (x*(5 + 6*x))

$$\frac{2 \left(2 - \frac{\left(12 x + 5\right)^{2}}{9 x \left(6 x + 5\right)}\right)}{\left(x \left(6 x + 5\right)\right)^{\frac{2}{3}}}$$

Simplify

The third derivative [src]

  /                 2 \           
  |     5*(5 + 12*x)  |           
2*|-4 + --------------|*(5 + 12*x)
  \     27*x*(5 + 6*x)/           
----------------------------------
                      5/3         
         (x*(5 + 6*x))

$$\frac{2 \left(-4 + \frac{5 \left(12 x + 5\right)^{2}}{27 x \left(6 x + 5\right)}\right) \left(12 x + 5\right)}{\left(x \left(6 x + 5\right)\right)^{\frac{5}{3}}}$$

Simplify

The graph

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Derivative of cbrt(6x^2+5x)

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The solution