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Identical expressions
y=(ln3x-2e^x)^ five
y equally (ln3x minus 2e to the power of x) to the power of 5
y equally (ln3x minus 2e to the power of x) to the power of five
y=(ln3x-2ex)5
y=ln3x-2ex5
y=(ln3x-2e^x)⁵
y=ln3x-2e^x^5
Similar expressions
y=(ln3x+2e^x)^5

The derivative of the function/ y=(ln3x-2e^x)^5

Derivative of y=(ln3x-2e^x)^5

The solution

You have entered [src]

                 5
/              x\ 
\log(3*x) - 2*E /

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

(log(3*x) - 2*exp(x))^5

Detail solution

Let $u = - 2 e^{x} + \log{\left(3 x \right)}$ .
Apply the power rule: $u^{5}$ goes to $5 u^{4}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(- 2 e^{x} + \log{\left(3 x \right)}\right)$ :
1. Differentiate $- 2 e^{x} + \log{\left(3 x \right)}$ term by term:
  1. Let $u = 3 x$ .
  2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x$ :
    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: $3$
    The result of the chain rule is:
    
    $\frac{1}{x}$
  4. The derivative of a constant times a function is the constant times the derivative of the function.
    1. The derivative of $e^{x}$ is itself.
    So, the result is: $- 2 e^{x}$
  The result is: $- 2 e^{x} + \frac{1}{x}$
The result of the chain rule is:

$5 \left(- 2 e^{x} + \log{\left(3 x \right)}\right)^{4} \left(- 2 e^{x} + \frac{1}{x}\right)$
Now simplify:

$\frac{\left(- 10 x e^{x} + 5\right) \left(2 e^{x} - \log{\left(3 x \right)}\right)^{4}}{x}$

The answer is:

$\frac{\left(- 10 x e^{x} + 5\right) \left(2 e^{x} - \log{\left(3 x \right)}\right)^{4}}{x}$

The graph

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The first derivative [src]

                 4              
/              x\  /      x   5\
\log(3*x) - 2*E / *|- 10*e  + -|
                   \          x/

$$\left(- 2 e^{x} + \log{\left(3 x \right)}\right)^{4} \left(- 10 e^{x} + \frac{5}{x}\right)$$

Simplify

The second derivative [src]

                     3 /              2                                 \
   /               x\  |  /  1      x\    /1       x\ /               x\|
-5*\-log(3*x) + 2*e / *|4*|- - + 2*e |  + |-- + 2*e |*\-log(3*x) + 2*e /|
                       |  \  x       /    | 2       |                   |
                       \                  \x        /                   /

$$- 5 \left(\left(2 e^{x} + \frac{1}{x^{2}}\right) \left(2 e^{x} - \log{\left(3 x \right)}\right) + 4 \left(2 e^{x} - \frac{1}{x}\right)^{2}\right) \left(2 e^{x} - \log{\left(3 x \right)}\right)^{3}$$

Simplify

The third derivative [src]

                      2 /              3                     2                                                            \
    /               x\  |  /  1      x\    /               x\  /  1     x\     /1       x\ /  1      x\ /               x\|
-10*\-log(3*x) + 2*e / *|6*|- - + 2*e |  + \-log(3*x) + 2*e / *|- -- + e | + 6*|-- + 2*e |*|- - + 2*e |*\-log(3*x) + 2*e /|
                        |  \  x       /                        |   3     |     | 2       | \  x       /                   |
                        \                                      \  x      /     \x        /                                /

$$- 10 \left(2 e^{x} - \log{\left(3 x \right)}\right)^{2} \left(\left(e^{x} - \frac{1}{x^{3}}\right) \left(2 e^{x} - \log{\left(3 x \right)}\right)^{2} + 6 \left(2 e^{x} + \frac{1}{x^{2}}\right) \left(2 e^{x} - \frac{1}{x}\right) \left(2 e^{x} - \log{\left(3 x \right)}\right) + 6 \left(2 e^{x} - \frac{1}{x}\right)^{3}\right)$$

Simplify

The graph

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Derivative of y=(ln3x-2e^x)^5

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Piecewise:

The solution