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Derivative of:
Derivative of e^-1
Derivative of 2*x^5
Derivative of f(x)=1
Derivative of (x+8)^2
Identical expressions
(five - four *x)/(three *x+ five)
(5 minus 4 multiply by x) divide by (3 multiply by x plus 5)
(five minus four multiply by x) divide by (three multiply by x plus five)
(5-4x)/(3x+5)
5-4x/3x+5
(5-4*x) divide by (3*x+5)
Similar expressions
(5+4*x)/(3*x+5)
(5-4*x)/(3*x-5)

The derivative of the function/ (5-4*x)/(3*x+5)

Derivative of (5-4x)/(3x+5)

The solution

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5 - 4*x
-------
3*x + 5

$$\frac{5 - 4 x}{3 x + 5}$$

(5 - 4*x)/(3*x + 5)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 5 - 4 x$ and $g{\left(x \right)} = 3 x + 5$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $5 - 4 x$ 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: $-4$
  The result is: $-4$
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:

$- \frac{35}{\left(3 x + 5\right)^{2}}$

The answer is:

$- \frac{35}{\left(3 x + 5\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     4      3*(5 - 4*x)
- ------- - -----------
  3*x + 5             2
             (3*x + 5)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

   /     3*(-5 + 4*x)\
54*|-4 + ------------|
   \       5 + 3*x   /
----------------------
               3      
      (5 + 3*x)

$$\frac{54 \left(-4 + \frac{3 \left(4 x - 5\right)}{3 x + 5}\right)}{\left(3 x + 5\right)^{3}}$$

Simplify