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Identical expressions
nine ^(sin(seven *x- two))
9 to the power of ( sinus of (7 multiply by x minus 2))
nine to the power of ( sinus of (seven multiply by x minus two))
9(sin(7*x-2))
9sin7*x-2
9^(sin(7x-2))
9(sin(7x-2))
9sin7x-2
9^sin7x-2
Similar expressions
9^(sin(7*x+2))

The derivative of the function/ 9^(sin(7*x-2))

Derivative of 9^(sin(7*x-2))

The solution

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 sin(7*x - 2)
9

$$9^{\sin{\left(7 x - 2 \right)}}$$

9^sin(7*x - 2)

Detail solution

Let $u = \sin{\left(7 x - 2 \right)}$ .
$\frac{d}{d u} 9^{u} = 9^{u} \log{\left(9 \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(7 x - 2 \right)}$ :
1. Let $u = 7 x - 2$ .
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 $\frac{d}{d x} \left(7 x - 2\right)$ :
  1. Differentiate $7 x - 2$ term by term:
    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: $7$
    2. The derivative of the constant $-2$ is zero.
    The result is: $7$
  The result of the chain rule is:
  
  $7 \cos{\left(7 x - 2 \right)}$
The result of the chain rule is:

$7 \cdot 9^{\sin{\left(7 x - 2 \right)}} \log{\left(9 \right)} \cos{\left(7 x - 2 \right)}$
Now simplify:

$14 \cdot 9^{\sin{\left(7 x - 2 \right)}} \log{\left(3 \right)} \cos{\left(7 x - 2 \right)}$

The answer is:

$14 \cdot 9^{\sin{\left(7 x - 2 \right)}} \log{\left(3 \right)} \cos{\left(7 x - 2 \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   sin(7*x - 2)                    
7*9            *cos(7*x - 2)*log(9)

$$7 \cdot 9^{\sin{\left(7 x - 2 \right)}} \log{\left(9 \right)} \cos{\left(7 x - 2 \right)}$$

Simplify

The second derivative [src]

    sin(-2 + 7*x) /                    2                 \       
49*9             *\-sin(-2 + 7*x) + cos (-2 + 7*x)*log(9)/*log(9)

$$49 \cdot 9^{\sin{\left(7 x - 2 \right)}} \left(- \sin{\left(7 x - 2 \right)} + \log{\left(9 \right)} \cos^{2}{\left(7 x - 2 \right)}\right) \log{\left(9 \right)}$$

Simplify

The third derivative [src]

     sin(-2 + 7*x) /        2              2                            \                     
343*9             *\-1 + cos (-2 + 7*x)*log (9) - 3*log(9)*sin(-2 + 7*x)/*cos(-2 + 7*x)*log(9)

$$343 \cdot 9^{\sin{\left(7 x - 2 \right)}} \left(- 3 \log{\left(9 \right)} \sin{\left(7 x - 2 \right)} + \log{\left(9 \right)}^{2} \cos^{2}{\left(7 x - 2 \right)} - 1\right) \log{\left(9 \right)} \cos{\left(7 x - 2 \right)}$$

Simplify

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Derivative of 9^(sin(7*x-2))

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The solution