Derivative of sqrt(a*cost)

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  __________
\/ a*cos(t)

$$\sqrt{a \cos{\left(t \right)}}$$

sqrt(a*cos(t))

Detail solution

Let $u = a \cos{\left(t \right)}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{\partial}{\partial t} a \cos{\left(t \right)}$ :
1. 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:
    
    $\frac{d}{d t} \cos{\left(t \right)} = - \sin{\left(t \right)}$
  So, the result is: $- a \sin{\left(t \right)}$
The result of the chain rule is:

$- \frac{a \sin{\left(t \right)}}{2 \sqrt{a \cos{\left(t \right)}}}$

The answer is:

$- \frac{a \sin{\left(t \right)}}{2 \sqrt{a \cos{\left(t \right)}}}$

The first derivative [src]

   __________        
-\/ a*cos(t) *sin(t) 
---------------------
       2*cos(t)

$$- \frac{\sqrt{a \cos{\left(t \right)}} \sin{\left(t \right)}}{2 \cos{\left(t \right)}}$$

Simplify

The second derivative [src]

              /       2   \ 
   __________ |    sin (t)| 
-\/ a*cos(t) *|2 + -------| 
              |       2   | 
              \    cos (t)/ 
----------------------------
             4

$$- \frac{\sqrt{a \cos{\left(t \right)}} \left(\frac{\sin^{2}{\left(t \right)}}{\cos^{2}{\left(t \right)}} + 2\right)}{4}$$

Simplify

The third derivative [src]

              /         2   \        
   __________ |    3*sin (t)|        
-\/ a*cos(t) *|2 + ---------|*sin(t) 
              |        2    |        
              \     cos (t) /        
-------------------------------------
               8*cos(t)

$$- \frac{\sqrt{a \cos{\left(t \right)}} \left(\frac{3 \sin^{2}{\left(t \right)}}{\cos^{2}{\left(t \right)}} + 2\right) \sin{\left(t \right)}}{8 \cos{\left(t \right)}}$$

Simplify

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Derivative of sqrt(a*cost)

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