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Derivative of:
Derivative of cos^3(4x)
Derivative of (3x+4)^2(x-5)^3
Derivative of ylny
Derivative of cos^3(2x)
Integral of d{x}:
cos^3(4x)
Identical expressions
cos^ three (4x)
co sinus of e of cubed (4x)
co sinus of e of to the power of three (4x)
cos3(4x)
cos34x
cos³(4x)
cos to the power of 3(4x)
cos^34x
Similar expressions
y=e^x+cos^34x
y=√sin^2(x)+√3*cos^3*4*x
cos^34x

The derivative of the function/ cos^3(4x)

Derivative of cos^3(4x)

The solution

You have entered [src]

   3     
cos (4*x)

$$\cos^{3}{\left(4 x \right)}$$

d /   3     \
--\cos (4*x)/
dx

$$\frac{d}{d x} \cos^{3}{\left(4 x \right)}$$

Transform

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Derivative of cos^3(4x)