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Derivative of:
Derivative of y=sin^2(4x)+1/2cos8x
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Identical expressions
y=sin^ two (4x)+ one /2cos8x
y equally sinus of squared (4x) plus 1 divide by 2 co sinus of e of 8x
y equally sinus of to the power of two (4x) plus one divide by 2 co sinus of e of 8x
y=sin2(4x)+1/2cos8x
y=sin24x+1/2cos8x
y=sin²(4x)+1/2cos8x
y=sin to the power of 2(4x)+1/2cos8x
y=sin^24x+1/2cos8x
y=sin^2(4x)+1 divide by 2cos8x
Similar expressions
y=sin^2(4x)-1/2cos8x

The derivative of the function/ y=sin^2(4x)+1/2cos8x

Derivative of y=sin^2(4x)+1/2cos8x

The solution

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   2        cos(8*x)
sin (4*x) + --------
               2

$$\sin^{2}{\left(4 x \right)} + \frac{\cos{\left(8 x \right)}}{2}$$

d /   2        cos(8*x)\
--|sin (4*x) + --------|
dx\               2    /

$$\frac{d}{d x} \left(\sin^{2}{\left(4 x \right)} + \frac{\cos{\left(8 x \right)}}{2}\right)$$

Transform

Detail solution

Differentiate $\sin^{2}{\left(4 x \right)} + \frac{\cos{\left(8 x \right)}}{2}$ term by term:
1. Let $u = \sin{\left(4 x \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(4 x \right)}$ :
  1. Let $u = 4 x$ .
  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 $\frac{d}{d x} 4 x$ :
    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: $4$
    The result of the chain rule is:
    
    $4 \cos{\left(4 x \right)}$
  The result of the chain rule is:
  
  $8 \sin{\left(4 x \right)} \cos{\left(4 x \right)}$
4. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = 8 x$ .
  2. The derivative of cosine is negative sine:
    
    $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 8 x$ :
    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: $8$
    The result of the chain rule is:
    
    $- 8 \sin{\left(8 x \right)}$
  So, the result is: $- 4 \sin{\left(8 x \right)}$
The result is: $8 \sin{\left(4 x \right)} \cos{\left(4 x \right)} - 4 \sin{\left(8 x \right)}$
Now simplify:

$0$

The answer is:

$0$

The first derivative [src]

-4*sin(8*x) + 8*cos(4*x)*sin(4*x)

$$8 \sin{\left(4 x \right)} \cos{\left(4 x \right)} - 4 \sin{\left(8 x \right)}$$

Simplify

The second derivative [src]

   /   2           2                \
32*\cos (4*x) - sin (4*x) - cos(8*x)/

$$32 \left(- \sin^{2}{\left(4 x \right)} + \cos^{2}{\left(4 x \right)} - \cos{\left(8 x \right)}\right)$$

Simplify

The third derivative [src]

256*(-2*cos(4*x)*sin(4*x) + sin(8*x))

$$256 \left(- 2 \sin{\left(4 x \right)} \cos{\left(4 x \right)} + \sin{\left(8 x \right)}\right)$$

Simplify

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Derivative of y=sin^2(4x)+1/2cos8x

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