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Derivative of 5x^2
Identical expressions
two cos^ four (x)-4x^2
2 co sinus of e of to the power of 4(x) minus 4x squared
two co sinus of e of to the power of four (x) minus 4x squared
2cos4(x)-4x2
2cos4x-4x2
2cos⁴(x)-4x²
2cos to the power of 4(x)-4x to the power of 2
2cos^4x-4x^2
Similar expressions
2cos^4(x)+4x^2

The derivative of the function/ 2cos^4(x)-4x^2

Derivative of 2cos^4(x)-4x^2

The solution

You have entered [src]

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

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

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

$$\frac{d}{d x} \left(- 4 x^{2} + 2 \cos^{4}{\left(x \right)}\right)$$

Transform

Detail solution

Differentiate $- 4 x^{2} + 2 \cos^{4}{\left(x \right)}$ 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: $u^{4}$ goes to $4 u^{3}$
  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:
    
    $- 4 \sin{\left(x \right)} \cos^{3}{\left(x \right)}$
  So, the result is: $- 8 \sin{\left(x \right)} \cos^{3}{\left(x \right)}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    So, the result is: $8 x$
  So, the result is: $- 8 x$
The result is: $- 8 x - 8 \sin{\left(x \right)} \cos^{3}{\left(x \right)}$

The answer is:

$- 8 x - 8 \sin{\left(x \right)} \cos^{3}{\left(x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

            3          
-8*x - 8*cos (x)*sin(x)

$$- 8 x - 8 \sin{\left(x \right)} \cos^{3}{\left(x \right)}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

   /       2           2   \              
16*\- 3*sin (x) + 5*cos (x)/*cos(x)*sin(x)

$$16 \left(- 3 \sin^{2}{\left(x \right)} + 5 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}$$

Simplify

The graph

Derivative of 2cos^4(x)-4x^2