Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of 2*x*sin(x)
Derivative of (x^2+3*x-10)^(5/2)
Derivative of (x+1)*(x+2)^2
Derivative of -x+1
Identical expressions
two exp(x- two)*ln(x-2)
2 exponent of (x minus 2) multiply by ln(x minus 2)
two exponent of (x minus two) multiply by ln(x minus 2)
2exp(x-2)ln(x-2)
2expx-2lnx-2
Similar expressions
2exp(x+2)*ln(x-2)
2exp(x-2)*ln(x+2)

The derivative of the function/ 2exp(x-2)*ln(x-2)

Derivative of 2exp(x-2)*ln(x-2)

The solution

You have entered [src]

   x - 2           
2*e     *log(x - 2)

$$2 e^{x - 2} \log{\left(x - 2 \right)}$$

(2*exp(x - 2))*log(x - 2)

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   x - 2                      
2*e           x - 2           
-------- + 2*e     *log(x - 2)
 x - 2

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of 2exp(x-2)*ln(x-2)

The graph:

Piecewise:

The solution