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The derivative of the function/ y=x+cos^2x

Derivative of y=x+cos^2x

The solution

You have entered [src]

       2   
x + cos (x)

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

x + cos(x)^2

Detail solution

Differentiate $x + \cos^{2}{\left(x \right)}$ term by term:
1. Apply the power rule: $x$ goes to $1$
2. Let $u = \cos{\left(x \right)}$ .
3. Apply the power rule: $u^{2}$ goes to $2 u$
4. 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:
  
  $- 2 \sin{\left(x \right)} \cos{\left(x \right)}$
The result is: $- 2 \sin{\left(x \right)} \cos{\left(x \right)} + 1$
Now simplify:

$1 - \sin{\left(2 x \right)}$

The answer is:

$1 - \sin{\left(2 x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

8*cos(x)*sin(x)

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

Simplify

The graph

Derivative of y=x+cos^2x