Derivative of 6^cos(3x)

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 cos(3*x)
6

$$6^{\cos{\left(3 x \right)}}$$

6^cos(3*x)

Detail solution

Let $u = \cos{\left(3 x \right)}$ .
$\frac{d}{d u} 6^{u} = 6^{u} \log{\left(6 \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(3 x \right)}$ :
1. Let $u = 3 x$ .
2. The derivative of cosine is negative sine:
  
  $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
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:
  
  $- 3 \sin{\left(3 x \right)}$
The result of the chain rule is:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    cos(3*x)                
-3*6        *log(6)*sin(3*x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

    cos(3*x) /       2       2                         \                
27*6        *\1 - log (6)*sin (3*x) + 3*cos(3*x)*log(6)/*log(6)*sin(3*x)

$$27 \cdot 6^{\cos{\left(3 x \right)}} \left(- \log{\left(6 \right)}^{2} \sin^{2}{\left(3 x \right)} + 3 \log{\left(6 \right)} \cos{\left(3 x \right)} + 1\right) \log{\left(6 \right)} \sin{\left(3 x \right)}$$

Simplify

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Identical expressions

Derivative of 6^cos(3x)

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The solution