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Derivative of y=sin5x-1/3sin³5x
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Similar expressions
y=sin5x+1/3sin³5x

The derivative of the function/ y=sin5x-1/3sin³5x

Derivative of y=sin5x-1/3sin³5x

The solution

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              35   
           sin  (x)
sin(5*x) - --------
              3

$$- \frac{\sin^{35}{\left(x \right)}}{3} + \sin{\left(5 x \right)}$$

  /              35   \
d |           sin  (x)|
--|sin(5*x) - --------|
dx\              3    /

$$\frac{d}{d x} \left(- \frac{\sin^{35}{\left(x \right)}}{3} + \sin{\left(5 x \right)}\right)$$

Transform

Detail solution

Differentiate $- \frac{\sin^{35}{\left(x \right)}}{3} + \sin{\left(5 x \right)}$ term by term:
1. Let $u = 5 x$ .
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 $\frac{d}{d x} 5 x$ :
  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: $5$
  The result of the chain rule is:
  
  $5 \cos{\left(5 x \right)}$
4. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Let $u = \sin{\left(x \right)}$ .
    2. Apply the power rule: $u^{35}$ goes to $35 u^{34}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
      1. The derivative of sine is cosine:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      The result of the chain rule is:
      
      $35 \sin^{34}{\left(x \right)} \cos{\left(x \right)}$
    So, the result is: $\frac{35 \sin^{34}{\left(x \right)} \cos{\left(x \right)}}{3}$
  So, the result is: $- \frac{35 \sin^{34}{\left(x \right)} \cos{\left(x \right)}}{3}$
The result is: $- \frac{35 \sin^{34}{\left(x \right)} \cos{\left(x \right)}}{3} + 5 \cos{\left(5 x \right)}$

The answer is:

$- \frac{35 \sin^{34}{\left(x \right)} \cos{\left(x \right)}}{3} + 5 \cos{\left(5 x \right)}$

The graph

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The first derivative [src]

                   34          
             35*sin  (x)*cos(x)
5*cos(5*x) - ------------------
                     3

$$- \frac{35 \sin^{34}{\left(x \right)} \cos{\left(x \right)}}{3} + 5 \cos{\left(5 x \right)}$$

Simplify

The second derivative [src]

  /                   35             2       33   \
  |              7*sin  (x)   238*cos (x)*sin  (x)|
5*|-5*sin(5*x) + ---------- - --------------------|
  \                  3                 3          /

$$5 \cdot \left(\frac{7 \sin^{35}{\left(x \right)}}{3} - \frac{238 \sin^{33}{\left(x \right)} \cos^{2}{\left(x \right)}}{3} - 5 \sin{\left(5 x \right)}\right)$$

Simplify

The third derivative [src]

  /                                              34          \
  |                       3       32      721*sin  (x)*cos(x)|
5*|-25*cos(5*x) - 2618*cos (x)*sin  (x) + -------------------|
  \                                                3         /

$$5 \cdot \left(\frac{721 \sin^{34}{\left(x \right)} \cos{\left(x \right)}}{3} - 2618 \sin^{32}{\left(x \right)} \cos^{3}{\left(x \right)} - 25 \cos{\left(5 x \right)}\right)$$

Simplify

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Derivative of y=sin5x-1/3sin³5x

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