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Derivative of:
Derivative of 4*x^(3/2)
Derivative of 3x-x^3
Derivative of (2*x-3)/(x+5)
Derivative of 2*x^3-4*x^2+x-1
Identical expressions
y=e^(2x- three)/(x+ five)
y equally e to the power of (2x minus 3) divide by (x plus 5)
y equally e to the power of (2x minus three) divide by (x plus five)
y=e(2x-3)/(x+5)
y=e2x-3/x+5
y=e^2x-3/x+5
y=e^(2x-3) divide by (x+5)
Similar expressions
y=e^(2x-3)/(x-5)
y=e^(2x+3)/(x+5)

The derivative of the function/ y=e^(2x-3)/(x+5)

Derivative of y=e^(2x-3)/(x+5)

The solution

You have entered [src]

 2*x - 3
E       
--------
 x + 5

$$\frac{e^{2 x - 3}}{x + 5}$$

E^(2*x - 3)/(x + 5)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = e^{2 x - 3}$ and $g{\left(x \right)} = x + 5$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 2 x - 3$ .
2. The derivative of $e^{u}$ is itself.
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x - 3\right)$ :
  1. Differentiate $2 x - 3$ term by term:
    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: $2$
    2. The derivative of the constant $-3$ is zero.
    The result is: $2$
  The result of the chain rule is:
  
  $2 e^{2 x - 3}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x + 5$ term by term:
  1. The derivative of the constant $5$ is zero.
  2. Apply the power rule: $x$ goes to $1$
  The result is: $1$
Now plug in to the quotient rule:

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

$\frac{\left(2 x + 9\right) e^{2 x - 3}}{\left(x + 5\right)^{2}}$

The answer is:

$\frac{\left(2 x + 9\right) e^{2 x - 3}}{\left(x + 5\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /       1         2  \  -3 + 2*x
2*|2 + -------- - -----|*e        
  |           2   5 + x|          
  \    (5 + x)         /          
----------------------------------
              5 + x

$$\frac{2 \left(2 - \frac{2}{x + 5} + \frac{1}{\left(x + 5\right)^{2}}\right) e^{2 x - 3}}{x + 5}$$

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of y=e^(2x-3)/(x+5)

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