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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

Plot the graph f(x)

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