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Derivative of:
Derivative of cos(5-3*x)
Derivative of cos^3(t)
Derivative of asin(y^2)
Derivative of (5x^2+2)×(x^2-4)^3
Identical expressions
((three *x)/(two *x- one))
((3 multiply by x) divide by (2 multiply by x minus 1))
((three multiply by x) divide by (two multiply by x minus one))
((3x)/(2x-1))
3x/2x-1
((3*x) divide by (2*x-1))
Similar expressions
((3*x)/(2*x+1))
y=(e^(1-3x))/(2x-1)

The derivative of the function/ ((3*x)/(2*x-1))

Derivative of ((3x)/(2x-1))

The solution

You have entered [src]

  3*x  
-------
2*x - 1

$$\frac{3 x}{2 x - 1}$$

(3*x)/(2*x - 1)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 3 x$ and $g{\left(x \right)} = 2 x - 1$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
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: $3$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $2 x - 1$ term by term:
  1. The derivative of the constant $-1$ 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: $2$
  The result is: $2$
Now plug in to the quotient rule:

$- \frac{3}{\left(2 x - 1\right)^{2}}$

The answer is:

$- \frac{3}{\left(2 x - 1\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   3         6*x    
------- - ----------
2*x - 1            2
          (2*x - 1)

$$- \frac{6 x}{\left(2 x - 1\right)^{2}} + \frac{3}{2 x - 1}$$

Simplify

The second derivative [src]

   /       2*x   \
12*|-1 + --------|
   \     -1 + 2*x/
------------------
             2    
   (-1 + 2*x)

$$\frac{12 \left(\frac{2 x}{2 x - 1} - 1\right)}{\left(2 x - 1\right)^{2}}$$

Simplify

The third derivative [src]

   /      2*x   \
72*|1 - --------|
   \    -1 + 2*x/
-----------------
             3   
   (-1 + 2*x)

$$\frac{72 \left(- \frac{2 x}{2 x - 1} + 1\right)}{\left(2 x - 1\right)^{3}}$$

Simplify