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two ^((x- three)sin(three *x))
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2((x-3)sin(3*x))
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Similar expressions
2^((x+3)sin(3*x))

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

Derivative of 2^((x-3)sin(3*x))

The solution

You have entered [src]

 (x - 3)*sin(3*x)
2

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

2^((x - 3)*sin(3*x))

Detail solution

Let $u = \left(x - 3\right) \sin{\left(3 x \right)}$ .
$\frac{d}{d u} 2^{u} = 2^{u} \log{\left(2 \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 3\right) \sin{\left(3 x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x - 3$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Differentiate $x - 3$ term by term:
    1. Apply the power rule: $x$ goes to $1$
    2. The derivative of the constant $-3$ is zero.
    The result is: $1$
  $g{\left(x \right)} = \sin{\left(3 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = 3 x$ .
  2. The derivative of sine is cosine:
    
    $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
  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:
    
    $3 \cos{\left(3 x \right)}$
  The result is: $3 \left(x - 3\right) \cos{\left(3 x \right)} + \sin{\left(3 x \right)}$
The result of the chain rule is:

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

$2^{\left(x - 3\right) \sin{\left(3 x \right)}} \left(\left(3 x - 9\right) \cos{\left(3 x \right)} + \sin{\left(3 x \right)}\right) \log{\left(2 \right)}$

The answer is:

$2^{\left(x - 3\right) \sin{\left(3 x \right)}} \left(\left(3 x - 9\right) \cos{\left(3 x \right)} + \sin{\left(3 x \right)}\right) \log{\left(2 \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 (x - 3)*sin(3*x)                                       
2                *(3*(x - 3)*cos(3*x) + sin(3*x))*log(2)

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

Simplify

The second derivative [src]

 (-3 + x)*sin(3*x) /                                             2                             \       
2                 *\6*cos(3*x) + (3*(-3 + x)*cos(3*x) + sin(3*x)) *log(2) - 9*(-3 + x)*sin(3*x)/*log(2)

$$2^{\left(x - 3\right) \sin{\left(3 x \right)}} \left(- 9 \left(x - 3\right) \sin{\left(3 x \right)} + \left(3 \left(x - 3\right) \cos{\left(3 x \right)} + \sin{\left(3 x \right)}\right)^{2} \log{\left(2 \right)} + 6 \cos{\left(3 x \right)}\right) \log{\left(2 \right)}$$

Simplify

The third derivative [src]

 (-3 + x)*sin(3*x) /                                               3    2                                                                                                          \       
2                 *\-27*sin(3*x) + (3*(-3 + x)*cos(3*x) + sin(3*x)) *log (2) - 27*(-3 + x)*cos(3*x) - 9*(-2*cos(3*x) + 3*(-3 + x)*sin(3*x))*(3*(-3 + x)*cos(3*x) + sin(3*x))*log(2)/*log(2)

$$2^{\left(x - 3\right) \sin{\left(3 x \right)}} \left(- 27 \left(x - 3\right) \cos{\left(3 x \right)} - 9 \left(3 \left(x - 3\right) \sin{\left(3 x \right)} - 2 \cos{\left(3 x \right)}\right) \left(3 \left(x - 3\right) \cos{\left(3 x \right)} + \sin{\left(3 x \right)}\right) \log{\left(2 \right)} + \left(3 \left(x - 3\right) \cos{\left(3 x \right)} + \sin{\left(3 x \right)}\right)^{3} \log{\left(2 \right)}^{2} - 27 \sin{\left(3 x \right)}\right) \log{\left(2 \right)}$$

Simplify

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Derivative of 2^((x-3)sin(3*x))

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The solution