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Identical expressions
ten ^(one -sin(three *x)^(four))
10 to the power of (1 minus sinus of (3 multiply by x) to the power of (4))
ten to the power of (one minus sinus of (three multiply by x) to the power of (four))
10(1-sin(3*x)(4))
101-sin3*x4
10^(1-sin(3x)^(4))
10(1-sin(3x)(4))
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Similar expressions
10^(1+sin(3*x)^(4))

The derivative of the function/ 10^(1-sin(3*x)^(4))

Derivative of 10^(1-sin(3*x)^(4))

The solution

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         4     
  1 - sin (3*x)
10

$$10^{1 - \sin^{4}{\left(3 x \right)}}$$

  /         4     \
d |  1 - sin (3*x)|
--\10             /
dx

$$\frac{d}{d x} 10^{1 - \sin^{4}{\left(3 x \right)}}$$

Transform

Detail solution

Let $u = 1 - \sin^{4}{\left(3 x \right)}$ .
$\frac{d}{d u} 10^{u} = 10^{u} \log{\left(10 \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(1 - \sin^{4}{\left(3 x \right)}\right)$ :
1. Differentiate $1 - \sin^{4}{\left(3 x \right)}$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Let $u = \sin{\left(3 x \right)}$ .
    2. Apply the power rule: $u^{4}$ goes to $4 u^{3}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(3 x \right)}$ :
      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 $\frac{d}{d x} 3 x$ :
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        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 of the chain rule is:
      
      $12 \sin^{3}{\left(3 x \right)} \cos{\left(3 x \right)}$
    So, the result is: $- 12 \sin^{3}{\left(3 x \right)} \cos{\left(3 x \right)}$
  The result is: $- 12 \sin^{3}{\left(3 x \right)} \cos{\left(3 x \right)}$
The result of the chain rule is:

$- 12 \cdot 10^{1 - \sin^{4}{\left(3 x \right)}} \log{\left(10 \right)} \sin^{3}{\left(3 x \right)} \cos{\left(3 x \right)}$
Now simplify:

$- 120 \cdot 10^{- \sin^{4}{\left(3 x \right)}} \log{\left(10 \right)} \sin^{3}{\left(3 x \right)} \cos{\left(3 x \right)}$

The answer is:

$- 120 \cdot 10^{- \sin^{4}{\left(3 x \right)}} \log{\left(10 \right)} \sin^{3}{\left(3 x \right)} \cos{\left(3 x \right)}$

The graph

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Plot the graph f'(x)

The first derivative [src]

             4                                
      1 - sin (3*x)    3                      
-12*10             *sin (3*x)*cos(3*x)*log(10)

$$- 12 \cdot 10^{1 - \sin^{4}{\left(3 x \right)}} \log{\left(10 \right)} \sin^{3}{\left(3 x \right)} \cos{\left(3 x \right)}$$

Simplify

The second derivative [src]

          4                                                                                 
      -sin (3*x)    2      /   2             2             2         4             \        
360*10          *sin (3*x)*\sin (3*x) - 3*cos (3*x) + 4*cos (3*x)*sin (3*x)*log(10)/*log(10)

$$360 \cdot 10^{- \sin^{4}{\left(3 x \right)}} \left(4 \log{\left(10 \right)} \sin^{4}{\left(3 x \right)} \cos^{2}{\left(3 x \right)} + \sin^{2}{\left(3 x \right)} - 3 \cos^{2}{\left(3 x \right)}\right) \log{\left(10 \right)} \sin^{2}{\left(3 x \right)}$$

Simplify

The third derivative [src]

           4                                                                                                                                                     
       -sin (3*x) /       2             2             6                     2         2        8              2         4             \                          
2160*10          *\- 3*cos (3*x) + 5*sin (3*x) - 6*sin (3*x)*log(10) - 8*cos (3*x)*log (10)*sin (3*x) + 18*cos (3*x)*sin (3*x)*log(10)/*cos(3*x)*log(10)*sin(3*x)

$$2160 \cdot 10^{- \sin^{4}{\left(3 x \right)}} \left(- 8 \log{\left(10 \right)}^{2} \sin^{8}{\left(3 x \right)} \cos^{2}{\left(3 x \right)} - 6 \log{\left(10 \right)} \sin^{6}{\left(3 x \right)} + 18 \log{\left(10 \right)} \sin^{4}{\left(3 x \right)} \cos^{2}{\left(3 x \right)} + 5 \sin^{2}{\left(3 x \right)} - 3 \cos^{2}{\left(3 x \right)}\right) \log{\left(10 \right)} \sin{\left(3 x \right)} \cos{\left(3 x \right)}$$

Simplify

The graph

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Derivative of 10^(1-sin(3*x)^(4))

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The solution