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Derivative of e^x^x
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Derivative of x^2*e^(3-2*x)*log(2)^x
Identical expressions
x^ two *e^(three - two *x)*log(two)^x
x squared multiply by e to the power of (3 minus 2 multiply by x) multiply by logarithm of (2) to the power of x
x to the power of two multiply by e to the power of (three minus two multiply by x) multiply by logarithm of (two) to the power of x
x2*e(3-2*x)*log(2)x
x2*e3-2*x*log2x
x²*e^(3-2*x)*log(2)^x
x to the power of 2*e to the power of (3-2*x)*log(2) to the power of x
x^2e^(3-2x)log(2)^x
x2e(3-2x)log(2)x
x2e3-2xlog2x
x^2e^3-2xlog2^x
Similar expressions
x^2*e^(3+2*x)*log(2)^x

The derivative of the function/ x^2*e^(3-2*x)*log(2)^x

Derivative of x^2e^(3-2x)*log(2)^x

The solution

You have entered [src]

 2  3 - 2*x    x   
x *E       *log (2)

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

(x^2*E^(3 - 2*x))*log(2)^x

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

   x    /             2    2    2                                   \  3 - 2*x
log (2)*\2 - 8*x + 4*x  + x *log (log(2)) - 4*x*(-1 + x)*log(log(2))/*e

$$\left(x^{2} \log{\left(\log{\left(2 \right)} \right)}^{2} + 4 x^{2} - 4 x \left(x - 1\right) \log{\left(\log{\left(2 \right)} \right)} - 8 x + 2\right) e^{3 - 2 x} \log{\left(2 \right)}^{x}$$

Simplify

The third derivative [src]

   x    /         2           2    3             /             2\                      2                 \  3 - 2*x
log (2)*\-12 - 8*x  + 24*x + x *log (log(2)) + 6*\1 - 4*x + 2*x /*log(log(2)) - 6*x*log (log(2))*(-1 + x)/*e

$$\left(- 8 x^{2} + x^{2} \log{\left(\log{\left(2 \right)} \right)}^{3} - 6 x \left(x - 1\right) \log{\left(\log{\left(2 \right)} \right)}^{2} + 24 x + 6 \left(2 x^{2} - 4 x + 1\right) \log{\left(\log{\left(2 \right)} \right)} - 12\right) e^{3 - 2 x} \log{\left(2 \right)}^{x}$$

Simplify

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Derivative of x^2e^(3-2x)*log(2)^x

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Piecewise:

The solution

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Derivative of x^2*e^(3-2*x)*log(2)^x

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Piecewise:

The solution

Derivative of x^2e^(3-2x)*log(2)^x