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The derivative of the function/ sin²x-cos²x

Derivative of sin²x-cos²x

The solution

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   2         2   
sin (x) - cos (x)

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

d /   2         2   \
--\sin (x) - cos (x)/
dx

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

Transform

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

4*cos(x)*sin(x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

-16*cos(x)*sin(x)

$$- 16 \sin{\left(x \right)} \cos{\left(x \right)}$$

Simplify

The graph

Derivative of sin²x-cos²x