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Identical expressions
y=sin(x)/ eight ^x
y equally sinus of (x) divide by 8 to the power of x
y equally sinus of (x) divide by eight to the power of x
y=sin(x)/8x
y=sinx/8x
y=sinx/8^x
y=sin(x) divide by 8^x
Similar expressions
y=sinx/8^x

The derivative of the function/ y=sin(x)/8^x

Derivative of y=sin(x)/8^x

The solution

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sin(x)
------
   x  
  8

$$\frac{\sin{\left(x \right)}}{8^{x}}$$

sin(x)/8^x

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = 8^{x}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. $\frac{d}{d x} 8^{x} = 8^{x} \log{\left(8 \right)}$
Now plug in to the quotient rule:

$8^{- 2 x} \left(- 8^{x} \log{\left(8 \right)} \sin{\left(x \right)} + 8^{x} \cos{\left(x \right)}\right)$
Now simplify:

$8^{- x} \left(- \log{\left(8^{\sin{\left(x \right)}} \right)} + \cos{\left(x \right)}\right)$

The answer is:

$8^{- x} \left(- \log{\left(8^{\sin{\left(x \right)}} \right)} + \cos{\left(x \right)}\right)$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 -x           -x              
8  *cos(x) - 8  *log(8)*sin(x)

$$- 8^{- x} \log{\left(8 \right)} \sin{\left(x \right)} + 8^{- x} \cos{\left(x \right)}$$

Simplify

The second derivative [src]

 -x /             2                            \
8  *\-sin(x) + log (8)*sin(x) - 2*cos(x)*log(8)/

$$8^{- x} \left(- \sin{\left(x \right)} + \log{\left(8 \right)}^{2} \sin{\left(x \right)} - 2 \log{\left(8 \right)} \cos{\left(x \right)}\right)$$

Simplify

The third derivative [src]

 -x /             3                  2                            \
8  *\-cos(x) - log (8)*sin(x) + 3*log (8)*cos(x) + 3*log(8)*sin(x)/

$$8^{- x} \left(- \log{\left(8 \right)}^{3} \sin{\left(x \right)} + 3 \log{\left(8 \right)} \sin{\left(x \right)} - \cos{\left(x \right)} + 3 \log{\left(8 \right)}^{2} \cos{\left(x \right)}\right)$$

Simplify

The graph

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Derivative of y=sin(x)/8^x

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The solution