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The derivative of the function/ sin(8x)^2

Derivative of sin(8x)^2

The solution

You have entered [src]

   2     
sin (8*x)

$$\sin^{2}{\left(8 x \right)}$$

sin(8*x)^2

Detail solution

Let $u = \sin{\left(8 x \right)}$ .
Apply the power rule: $u^{2}$ goes to $2 u$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(8 x \right)}$ :
1. Let $u = 8 x$ .
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 $\frac{d}{d x} 8 x$ :
  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:
  
  $8 \cos{\left(8 x \right)}$
The result of the chain rule is:

$16 \sin{\left(8 x \right)} \cos{\left(8 x \right)}$

The answer is:

$16 \sin{\left(8 x \right)} \cos{\left(8 x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

16*cos(8*x)*sin(8*x)

$$16 \sin{\left(8 x \right)} \cos{\left(8 x \right)}$$

Simplify

The second derivative [src]

    /   2           2     \
128*\cos (8*x) - sin (8*x)/

$$128 \left(- \sin^{2}{\left(8 x \right)} + \cos^{2}{\left(8 x \right)}\right)$$

Simplify

The third derivative [src]

-4096*cos(8*x)*sin(8*x)

$$- 4096 \sin{\left(8 x \right)} \cos{\left(8 x \right)}$$

Simplify