Mister Exam

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The derivative of the function/ x*a^x

Derivative of x*a^x

The solution

You have entered [src]

   x
x*a

$$a^{x} x$$

x*a^x

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
$g{\left(x \right)} = a^{x}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. $\frac{\partial}{\partial x} a^{x} = a^{x} \log{\left(a \right)}$
The result is: $a^{x} x \log{\left(a \right)} + a^{x}$
Now simplify:

$a^{x} \left(x \log{\left(a \right)} + 1\right)$

The answer is:

$a^{x} \left(x \log{\left(a \right)} + 1\right)$

The first derivative [src]

 x      x       
a  + x*a *log(a)

$$a^{x} x \log{\left(a \right)} + a^{x}$$

Simplify

The second derivative [src]

 x                      
a *(2 + x*log(a))*log(a)

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

Simplify

The third derivative [src]

 x    2                  
a *log (a)*(3 + x*log(a))

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

Simplify