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Identical expressions
(sin(3x))^ five *ln(7x)-10x
( sinus of (3x)) to the power of 5 multiply by ln(7x) minus 10x
( sinus of (3x)) to the power of five multiply by ln(7x) minus 10x
(sin(3x))5*ln(7x)-10x
sin3x5*ln7x-10x
(sin(3x))⁵*ln(7x)-10x
(sin(3x))^5ln(7x)-10x
(sin(3x))5ln(7x)-10x
sin3x5ln7x-10x
sin3x^5ln7x-10x
Similar expressions
(sin(3x))^5*ln(7x)+10x

The derivative of the function/ (sin(3x))^5*ln(7x)-10x

Derivative of (sin(3x))^5*ln(7x)-10x

The solution

You have entered [src]

   5                     
sin (3*x)*log(7*x) - 10*x

$$- 10 x + \log{\left(7 x \right)} \sin^{5}{\left(3 x \right)}$$

sin(3*x)^5*log(7*x) - 10*x

Detail solution

Differentiate $- 10 x + \log{\left(7 x \right)} \sin^{5}{\left(3 x \right)}$ term by term:
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \sin^{5}{\left(3 x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = \sin{\left(3 x \right)}$ .
  2. Apply the power rule: $u^{5}$ goes to $5 u^{4}$
  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 $\frac{d}{d x} 3 x$ :
      1. 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:
    
    $15 \sin^{4}{\left(3 x \right)} \cos{\left(3 x \right)}$
  $g{\left(x \right)} = \log{\left(7 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = 7 x$ .
  2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 7 x$ :
    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: $7$
    The result of the chain rule is:
    
    $\frac{1}{x}$
  The result is: $15 \log{\left(7 x \right)} \sin^{4}{\left(3 x \right)} \cos{\left(3 x \right)} + \frac{\sin^{5}{\left(3 x \right)}}{x}$
2. 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: $-10$
The result is: $15 \log{\left(7 x \right)} \sin^{4}{\left(3 x \right)} \cos{\left(3 x \right)} - 10 + \frac{\sin^{5}{\left(3 x \right)}}{x}$

The answer is:

$15 \log{\left(7 x \right)} \sin^{4}{\left(3 x \right)} \cos{\left(3 x \right)} - 10 + \frac{\sin^{5}{\left(3 x \right)}}{x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

         5                                      
      sin (3*x)         4                       
-10 + --------- + 15*sin (3*x)*cos(3*x)*log(7*x)
          x

$$15 \log{\left(7 x \right)} \sin^{4}{\left(3 x \right)} \cos{\left(3 x \right)} - 10 + \frac{\sin^{5}{\left(3 x \right)}}{x}$$

Simplify

The second derivative [src]

          /     2                                                                             \
   3      |  sin (3*x)         2                        2                 30*cos(3*x)*sin(3*x)|
sin (3*x)*|- --------- - 45*sin (3*x)*log(7*x) + 180*cos (3*x)*log(7*x) + --------------------|
          |       2                                                                x          |
          \      x                                                                            /

$$\left(- 45 \log{\left(7 x \right)} \sin^{2}{\left(3 x \right)} + 180 \log{\left(7 x \right)} \cos^{2}{\left(3 x \right)} + \frac{30 \sin{\left(3 x \right)} \cos{\left(3 x \right)}}{x} - \frac{\sin^{2}{\left(3 x \right)}}{x^{2}}\right) \sin^{3}{\left(3 x \right)}$$

Simplify

3-я производная [src]

          /         3             3                                                                           2                        2              \
   2      |  135*sin (3*x)   2*sin (3*x)           3                         2                          45*sin (3*x)*cos(3*x)   540*cos (3*x)*sin(3*x)|
sin (3*x)*|- ------------- + ----------- + 1620*cos (3*x)*log(7*x) - 1755*sin (3*x)*cos(3*x)*log(7*x) - --------------------- + ----------------------|
          |        x               3                                                                               2                      x           |
          \                       x                                                                               x                                   /

$$\left(- 1755 \log{\left(7 x \right)} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)} + 1620 \log{\left(7 x \right)} \cos^{3}{\left(3 x \right)} - \frac{135 \sin^{3}{\left(3 x \right)}}{x} + \frac{540 \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{x} - \frac{45 \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{x^{2}} + \frac{2 \sin^{3}{\left(3 x \right)}}{x^{3}}\right) \sin^{2}{\left(3 x \right)}$$

Simplify

The third derivative [src]

          /         3             3                                                                           2                        2              \
   2      |  135*sin (3*x)   2*sin (3*x)           3                         2                          45*sin (3*x)*cos(3*x)   540*cos (3*x)*sin(3*x)|
sin (3*x)*|- ------------- + ----------- + 1620*cos (3*x)*log(7*x) - 1755*sin (3*x)*cos(3*x)*log(7*x) - --------------------- + ----------------------|
          |        x               3                                                                               2                      x           |
          \                       x                                                                               x                                   /

Simplify

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Derivative of (sin(3x))^5*ln(7x)-10x

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