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five ^x*sin(x)
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five to the power of x multiply by sinus of (x)
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Similar expressions
5^x*sinx

The derivative of the function/ 5^x*sin(x)

Derivative of 5^x*sin(x)

The solution

You have entered [src]

 x       
5 *sin(x)

$$5^{x} \sin{\left(x \right)}$$

5^x*sin(x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = 5^{x}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. $\frac{d}{d x} 5^{x} = 5^{x} \log{\left(5 \right)}$
$g{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
The result is: $5^{x} \log{\left(5 \right)} \sin{\left(x \right)} + 5^{x} \cos{\left(x \right)}$
Now simplify:

$5^{x} \left(\log{\left(5 \right)} \sin{\left(x \right)} + \cos{\left(x \right)}\right)$

The answer is:

$5^{x} \left(\log{\left(5 \right)} \sin{\left(x \right)} + \cos{\left(x \right)}\right)$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 x           x              
5 *cos(x) + 5 *log(5)*sin(x)

$$5^{x} \log{\left(5 \right)} \sin{\left(x \right)} + 5^{x} \cos{\left(x \right)}$$

Simplify

The second derivative [src]

 x /             2                            \
5 *\-sin(x) + log (5)*sin(x) + 2*cos(x)*log(5)/

$$5^{x} \left(- \sin{\left(x \right)} + \log{\left(5 \right)}^{2} \sin{\left(x \right)} + 2 \log{\left(5 \right)} \cos{\left(x \right)}\right)$$

Simplify

The third derivative [src]

 x /             3                                    2          \
5 *\-cos(x) + log (5)*sin(x) - 3*log(5)*sin(x) + 3*log (5)*cos(x)/

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

Simplify

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Derivative of 5^x*sin(x)

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The solution