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Limit of the function:
sin(5*x)/x^2
Identical expressions
sin(five *x)/x^ two
sinus of (5 multiply by x) divide by x squared
sinus of (five multiply by x) divide by x to the power of two
sin(5*x)/x2
sin5*x/x2
sin(5*x)/x²
sin(5*x)/x to the power of 2
sin(5x)/x^2
sin(5x)/x2
sin5x/x2
sin5x/x^2
sin(5*x) divide by x^2

The derivative of the function/ sin(5*x)/x^2

Derivative of sin(5*x)/x^2

The solution

You have entered [src]

sin(5*x)
--------
    2   
   x

$$\frac{\sin{\left(5 x \right)}}{x^{2}}$$

sin(5*x)/x^2

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sin{\left(5 x \right)}$ and $g{\left(x \right)} = x^{2}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 5 x$ .
2. The derivative of sine is cosine:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 5 x$ :
  1. 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 of the chain rule is:
  
  $5 \cos{\left(5 x \right)}$
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:

$\frac{5 x^{2} \cos{\left(5 x \right)} - 2 x \sin{\left(5 x \right)}}{x^{4}}$
Now simplify:

$\frac{5 x \cos{\left(5 x \right)} - 2 \sin{\left(5 x \right)}}{x^{3}}$

The answer is:

$\frac{5 x \cos{\left(5 x \right)} - 2 \sin{\left(5 x \right)}}{x^{3}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  2*sin(5*x)   5*cos(5*x)
- ---------- + ----------
       3            2    
      x            x

$$\frac{5 \cos{\left(5 x \right)}}{x^{2}} - \frac{2 \sin{\left(5 x \right)}}{x^{3}}$$

Simplify

The second derivative [src]

               20*cos(5*x)   6*sin(5*x)
-25*sin(5*x) - ----------- + ----------
                    x             2    
                                 x     
---------------------------------------
                    2                  
                   x

$$\frac{- 25 \sin{\left(5 x \right)} - \frac{20 \cos{\left(5 x \right)}}{x} + \frac{6 \sin{\left(5 x \right)}}{x^{2}}}{x^{2}}$$

Simplify

The third derivative [src]

                24*sin(5*x)   90*cos(5*x)   150*sin(5*x)
-125*cos(5*x) - ----------- + ----------- + ------------
                      3             2            x      
                     x             x                    
--------------------------------------------------------
                            2                           
                           x

$$\frac{- 125 \cos{\left(5 x \right)} + \frac{150 \sin{\left(5 x \right)}}{x} + \frac{90 \cos{\left(5 x \right)}}{x^{2}} - \frac{24 \sin{\left(5 x \right)}}{x^{3}}}{x^{2}}$$

Simplify

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Identical expressions

Derivative of sin(5*x)/x^2

The graph:

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The solution