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Derivative of:
Derivative of 4*e^(2*x)
Derivative of -(3*pi*cos(pi*t/6))/2
Derivative of (2x-1)^3
Derivative of (1-2*x)^3
Identical expressions
sin(four *x)^(two)
sinus of (4 multiply by x) to the power of (2)
sinus of (four multiply by x) to the power of (two)
sin(4*x)(2)
sin4*x2
sin(4x)^(2)
sin(4x)(2)
sin4x2
sin4x^2

The derivative of the function/ sin(4*x)^(2)

Derivative of sin(4*x)^(2)

The solution

You have entered [src]

   2     
sin (4*x)

$$\sin^{2}{\left(4 x \right)}$$

d /   2     \
--\sin (4*x)/
dx

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

Transform

Detail solution

Let $u = \sin{\left(4 x \right)}$ .
Apply the power rule: $u^{2}$ goes to $2 u$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(4 x \right)}$ :
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:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

-512*cos(4*x)*sin(4*x)

$$- 512 \sin{\left(4 x \right)} \cos{\left(4 x \right)}$$

Simplify

The graph

Derivative of sin(4*x)^(2)