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Identical expressions
y=cos(x)/(three *x+ five)
y equally co sinus of e of (x) divide by (3 multiply by x plus 5)
y equally co sinus of e of (x) divide by (three multiply by x plus five)
y=cos(x)/(3x+5)
y=cosx/3x+5
y=cos(x) divide by (3*x+5)
Similar expressions
y=cos(x)/(3*x-5)
y=cosx/(3*x+5)

The derivative of the function/ y=cos(x)/(3*x+5)

Derivative of y=cos(x)/(3*x+5)

The solution

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

$$\frac{\cos{\left(x \right)}}{3 x + 5}$$

cos(x)/(3*x + 5)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \cos{\left(x \right)}$ and $g{\left(x \right)} = 3 x + 5$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of cosine is negative sine:
  
  $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $3 x + 5$ term by term:
  1. The derivative of the constant $5$ is zero.
  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: $3$
  The result is: $3$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

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The first derivative [src]

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

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

Simplify

The second derivative [src]

          6*sin(x)   18*cos(x) 
-cos(x) + -------- + ----------
          5 + 3*x             2
                     (5 + 3*x) 
-------------------------------
            5 + 3*x

$$\frac{- \cos{\left(x \right)} + \frac{6 \sin{\left(x \right)}}{3 x + 5} + \frac{18 \cos{\left(x \right)}}{\left(3 x + 5\right)^{2}}}{3 x + 5}$$

Simplify

The third derivative [src]

  162*cos(x)   54*sin(x)    9*cos(x)         
- ---------- - ---------- + -------- + sin(x)
           3            2   5 + 3*x          
  (5 + 3*x)    (5 + 3*x)                     
---------------------------------------------
                   5 + 3*x

$$\frac{\sin{\left(x \right)} + \frac{9 \cos{\left(x \right)}}{3 x + 5} - \frac{54 \sin{\left(x \right)}}{\left(3 x + 5\right)^{2}} - \frac{162 \cos{\left(x \right)}}{\left(3 x + 5\right)^{3}}}{3 x + 5}$$

Simplify

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Derivative of y=cos(x)/(3*x+5)

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The solution