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Identical expressions
(sin(x)/((x+ ten)^ one / three))
( sinus of (x) divide by ((x plus 10) to the power of 1 divide by 3))
( sinus of (x) divide by ((x plus ten) to the power of one divide by three))
(sin(x)/((x+10)1/3))
sinx/x+101/3
sinx/x+10^1/3
(sin(x) divide by ((x+10)^1 divide by 3))
Similar expressions
(sin(x)/((x-10)^1/3))
(sinx/((x+10)^1/3))

The derivative of the function/ (sin(x)/((x+10)^1/3))

Derivative of (sin(x)/((x+10)^1/3))

The solution

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  sin(x)  
----------
3 ________
\/ x + 10

$$\frac{\sin{\left(x \right)}}{\sqrt[3]{x + 10}}$$

sin(x)/(x + 10)^(1/3)

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = \sqrt[3]{x + 10}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x + 10$ .
2. Apply the power rule: $\sqrt[3]{u}$ goes to $\frac{1}{3 u^{\frac{2}{3}}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 10\right)$ :
  1. Differentiate $x + 10$ term by term:
    1. The derivative of the constant $10$ is zero.
    2. Apply the power rule: $x$ goes to $1$
    The result is: $1$
  The result of the chain rule is:
  
  $\frac{1}{3 \left(x + 10\right)^{\frac{2}{3}}}$
Now plug in to the quotient rule:

$\frac{\sqrt[3]{x + 10} \cos{\left(x \right)} - \frac{\sin{\left(x \right)}}{3 \left(x + 10\right)^{\frac{2}{3}}}}{\left(x + 10\right)^{\frac{2}{3}}}$
Now simplify:

$\frac{\left(3 x + 30\right) \cos{\left(x \right)} - \sin{\left(x \right)}}{3 \left(x + 10\right)^{\frac{4}{3}}}$

The answer is:

$\frac{\left(3 x + 30\right) \cos{\left(x \right)} - \sin{\left(x \right)}}{3 \left(x + 10\right)^{\frac{4}{3}}}$

The first derivative [src]

  cos(x)         sin(x)   
---------- - -------------
3 ________             4/3
\/ x + 10    3*(x + 10)

$$\frac{\cos{\left(x \right)}}{\sqrt[3]{x + 10}} - \frac{\sin{\left(x \right)}}{3 \left(x + 10\right)^{\frac{4}{3}}}$$

Simplify

The second derivative [src]

           2*cos(x)      4*sin(x) 
-sin(x) - ---------- + -----------
          3*(10 + x)             2
                       9*(10 + x) 
----------------------------------
            3 ________            
            \/ 10 + x

$$\frac{- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{3 \left(x + 10\right)} + \frac{4 \sin{\left(x \right)}}{9 \left(x + 10\right)^{2}}}{\sqrt[3]{x + 10}}$$

Simplify

The third derivative [src]

          sin(x)    28*sin(x)       4*cos(x) 
-cos(x) + ------ - ------------ + -----------
          10 + x              3             2
                   27*(10 + x)    3*(10 + x) 
---------------------------------------------
                  3 ________                 
                  \/ 10 + x

$$\frac{- \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x + 10} + \frac{4 \cos{\left(x \right)}}{3 \left(x + 10\right)^{2}} - \frac{28 \sin{\left(x \right)}}{27 \left(x + 10\right)^{3}}}{\sqrt[3]{x + 10}}$$

Simplify

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Derivative of (sin(x)/((x+10)^1/3))

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