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Derivative of:
Derivative of sin(x)^2
Derivative of e^(2-x)
Derivative of cos(x/3)
Derivative of √x
Identical expressions
(sinx/x^ two)+(x^(two / three))*cosx
( sinus of x divide by x squared ) plus (x to the power of (2 divide by 3)) multiply by co sinus of e of x
( sinus of x divide by x to the power of two) plus (x to the power of (two divide by three)) multiply by co sinus of e of x
(sinx/x2)+(x(2/3))*cosx
sinx/x2+x2/3*cosx
(sinx/x²)+(x^(2/3))*cosx
(sinx/x to the power of 2)+(x to the power of (2/3))*cosx
(sinx/x^2)+(x^(2/3))cosx
(sinx/x2)+(x(2/3))cosx
sinx/x2+x2/3cosx
sinx/x^2+x^2/3cosx
(sinx divide by x^2)+(x^(2 divide by 3))*cosx
Similar expressions
(sinx/x^2)-(x^(2/3))*cosx

The derivative of the function/ (sinx/x^2)+(x^(2/3))*cosx

Derivative of (sinx/x^2)+(x^(2/3))*cosx

The solution

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sin(x)    2/3       
------ + x   *cos(x)
   2                
  x

$$x^{\frac{2}{3}} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x^{2}}$$

d /sin(x)    2/3       \
--|------ + x   *cos(x)|
dx|   2                |
  \  x                 /

$$\frac{d}{d x} \left(x^{\frac{2}{3}} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x^{2}}\right)$$

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Detail solution

Differentiate $x^{\frac{2}{3}} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x^{2}}$ term by term:
1. 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 $g{\left(x \right)} = x^{2}$ .
  
  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. Apply the power rule: $x^{2}$ goes to $2 x$
  Now plug in to the quotient rule:
  
  $\frac{x^{2} \cos{\left(x \right)} - 2 x \sin{\left(x \right)}}{x^{4}}$
2. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x^{\frac{2}{3}}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x^{\frac{2}{3}}$ goes to $\frac{2}{3 \sqrt[3]{x}}$
  $g{\left(x \right)} = \cos{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  The result is: $- x^{\frac{2}{3}} \sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{3 \sqrt[3]{x}}$
The result is: $- x^{\frac{2}{3}} \sin{\left(x \right)} + \frac{x^{2} \cos{\left(x \right)} - 2 x \sin{\left(x \right)}}{x^{4}} + \frac{2 \cos{\left(x \right)}}{3 \sqrt[3]{x}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

cos(x)    2/3          2*sin(x)   2*cos(x)
------ - x   *sin(x) - -------- + --------
   2                       3        3 ___ 
  x                       x       3*\/ x

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

Simplify

The second derivative [src]

  sin(x)    2/3          4*cos(x)   6*sin(x)   4*sin(x)   2*cos(x)
- ------ - x   *cos(x) - -------- + -------- - -------- - --------
     2                       3          4        3 ___        4/3 
    x                       x          x       3*\/ x      9*x

$$- x^{\frac{2}{3}} \cos{\left(x \right)} - \frac{\sin{\left(x \right)}}{x^{2}} - \frac{4 \cos{\left(x \right)}}{x^{3}} + \frac{6 \sin{\left(x \right)}}{x^{4}} - \frac{4 \sin{\left(x \right)}}{3 \sqrt[3]{x}} - \frac{2 \cos{\left(x \right)}}{9 x^{\frac{4}{3}}}$$

Simplify

The third derivative [src]

 2/3          cos(x)   24*sin(x)   2*cos(x)   6*sin(x)   18*cos(x)   2*sin(x)   8*cos(x)
x   *sin(x) - ------ - --------- - -------- + -------- + --------- + -------- + --------
                 2          5       3 ___         3           4          4/3        7/3 
                x          x        \/ x         x           x        3*x       27*x

$$x^{\frac{2}{3}} \sin{\left(x \right)} - \frac{\cos{\left(x \right)}}{x^{2}} + \frac{6 \sin{\left(x \right)}}{x^{3}} + \frac{18 \cos{\left(x \right)}}{x^{4}} - \frac{24 \sin{\left(x \right)}}{x^{5}} - \frac{2 \cos{\left(x \right)}}{\sqrt[3]{x}} + \frac{2 \sin{\left(x \right)}}{3 x^{\frac{4}{3}}} + \frac{8 \cos{\left(x \right)}}{27 x^{\frac{7}{3}}}$$

Simplify

The graph

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Derivative of (sinx/x^2)+(x^(2/3))*cosx

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The solution