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Derivative of:
Derivative of y=sin^3(4x−1).
Derivative of y=tg2x
Derivative of y=sin^2(4x)+1/2cos8x
Derivative of y³
Identical expressions
y=sin^ three (4x− one).
y equally sinus of cubed (4x−1).
y equally sinus of to the power of three (4x− one).
y=sin3(4x−1).
y=sin34x−1.
y=sin³(4x−1).
y=sin to the power of 3(4x−1).
y=sin^34x−1.

The derivative of the function/ y=sin^3(4x−1).

Derivative of y=sin^3(4x−1).

The solution

You have entered [src]

   3         
sin (4*x - 1)

$$\sin^{3}{\left(4 x - 1 \right)}$$

d /   3         \
--\sin (4*x - 1)/
dx

$$\frac{d}{d x} \sin^{3}{\left(4 x - 1 \right)}$$

Transform

Detail solution

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

$12 \sin^{2}{\left(4 x - 1 \right)} \cos{\left(4 x - 1 \right)}$
Now simplify:

$12 \sin^{2}{\left(4 x - 1 \right)} \cos{\left(4 x - 1 \right)}$

The answer is:

$12 \sin^{2}{\left(4 x - 1 \right)} \cos{\left(4 x - 1 \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      2                      
12*sin (4*x - 1)*cos(4*x - 1)

$$12 \sin^{2}{\left(4 x - 1 \right)} \cos{\left(4 x - 1 \right)}$$

Simplify

The second derivative [src]

   /     2                  2          \              
48*\- sin (-1 + 4*x) + 2*cos (-1 + 4*x)/*sin(-1 + 4*x)

$$48 \left(- \sin^{2}{\left(4 x - 1 \right)} + 2 \cos^{2}{\left(4 x - 1 \right)}\right) \sin{\left(4 x - 1 \right)}$$

Simplify

The third derivative [src]

    /       2                  2          \              
192*\- 7*sin (-1 + 4*x) + 2*cos (-1 + 4*x)/*cos(-1 + 4*x)

$$192 \left(- 7 \sin^{2}{\left(4 x - 1 \right)} + 2 \cos^{2}{\left(4 x - 1 \right)}\right) \cos{\left(4 x - 1 \right)}$$

Simplify

The graph

Derivative of y=sin^3(4x−1).