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Derivative of:
Derivative of sqrt(x)+2
Derivative of log(x^2*e^x)
Derivative of log(x^2+5*sqrt(x)-3)
Derivative of e^x*log(x)
Identical expressions
log(x^ two *e^x)
logarithm of (x squared multiply by e to the power of x)
logarithm of (x to the power of two multiply by e to the power of x)
log(x2*ex)
logx2*ex
log(x²*e^x)
log(x to the power of 2*e to the power of x)
log(x^2e^x)
log(x2ex)
logx2ex
logx^2e^x

The derivative of the function/ log(x^2*e^x)

Derivative of log(x^2*e^x)

The solution

You have entered [src]

   / 2  x\
log\x *E /

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

log(x^2*E^x)

Detail solution

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

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

$\frac{x + 2}{x}$

The answer is:

$\frac{x + 2}{x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

/ 2  x        x\  -x
\x *e  + 2*x*e /*e  
--------------------
          2         
         x

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

             2           /     2      \     /     2      \                        
        6 + x  + 6*x   4*\2 + x  + 4*x/   2*\2 + x  + 4*x/   4*(2 + x)   6*(2 + x)
2 + x + ------------ - ---------------- - ---------------- + --------- + ---------
             x                 2                 x               x            2   
                              x                                              x    
----------------------------------------------------------------------------------
                                        x

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

Simplify

The graph

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Identical expressions

Derivative of log(x^2*e^x)

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The solution