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Derivative of:
Derivative of x^e
Derivative of (x^3-1)(x^2+x+1)
Derivative of e^(sin(1-3*x)^(2))
Derivative of -3*x^3+2*x^2-x-5
Canonical form:
cos(x)-sqrt(x+2)-1
Identical expressions
cos(x)-sqrt(x+ two)- one
co sinus of e of (x) minus square root of (x plus 2) minus 1
co sinus of e of (x) minus square root of (x plus two) minus one
cos(x)-√(x+2)-1
cosx-sqrtx+2-1
Similar expressions
cos(x)+sqrt(x+2)-1
cos(x)-sqrt(x+2)+1
cos(x)-sqrt(x-2)-1
cosx-sqrt(x+2)-1

The derivative of the function/ cos(x)-sqrt(x+2)-1

Derivative of cos(x)-sqrt(x+2)-1

The solution

You have entered [src]

           _______    
cos(x) - \/ x + 2  - 1

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

cos(x) - sqrt(x + 2) - 1

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

               1     
-sin(x) - -----------
              _______
          2*\/ x + 2

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

       3               
- ------------ + sin(x)
           5/2         
  8*(2 + x)

$$\sin{\left(x \right)} - \frac{3}{8 \left(x + 2\right)^{\frac{5}{2}}}$$

Simplify