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Derivative of:
Derivative of sinx/x^2
Derivative of x^2-1
Derivative of sin8x
Derivative of sin^4xcos^4x
Identical expressions
4cos^ three (x)-3cos(x)
4 co sinus of e of cubed (x) minus 3 co sinus of e of (x)
4 co sinus of e of to the power of three (x) minus 3 co sinus of e of (x)
4cos3(x)-3cos(x)
4cos3x-3cosx
4cos³(x)-3cos(x)
4cos to the power of 3(x)-3cos(x)
4cos^3x-3cosx
Similar expressions
4cos^3(x)+3cos(x)
4cos^3(x)-3cosx

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

Derivative of 4cos^3(x)-3cos(x)

The solution

You have entered [src]

     3              
4*cos (x) - 3*cos(x)

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

4*cos(x)^3 - 3*cos(x)

Detail solution

Differentiate $4 \cos^{3}{\left(x \right)} - 3 \cos{\left(x \right)}$ term by term:
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \cos{\left(x \right)}$ .
  2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    The result of the chain rule is:
    
    $- 3 \sin{\left(x \right)} \cos^{2}{\left(x \right)}$
  So, the result is: $- 12 \sin{\left(x \right)} \cos^{2}{\left(x \right)}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  So, the result is: $3 \sin{\left(x \right)}$
The result is: $- 12 \sin{\left(x \right)} \cos^{2}{\left(x \right)} + 3 \sin{\left(x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

$$- 12 \sin{\left(x \right)} \cos^{2}{\left(x \right)} + 3 \sin{\left(x \right)}$$

Simplify

The second derivative [src]

  /         2           2   \       
3*\1 - 4*cos (x) + 8*sin (x)/*cos(x)

$$3 \left(8 \sin^{2}{\left(x \right)} - 4 \cos^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)}$$

Simplify

The third derivative [src]

  /          2            2   \       
3*\-1 - 8*sin (x) + 28*cos (x)/*sin(x)

$$3 \left(- 8 \sin^{2}{\left(x \right)} + 28 \cos^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)}$$

Simplify