Derivative of √3sin(x)+cos(x)

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  __________         
\/ 3*sin(x)  + cos(x)

$$\sqrt{3 \sin{\left(x \right)}} + \cos{\left(x \right)}$$

sqrt(3*sin(x)) + cos(x)

Detail solution

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

The answer is:

$- \sin{\left(x \right)} + \frac{\sqrt{3} \cos{\left(x \right)}}{2 \sqrt{\sin{\left(x \right)}}}$

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The first derivative [src]

            ___   ________       
          \/ 3 *\/ sin(x) *cos(x)
-sin(x) + -----------------------
                  2*sin(x)

$$\frac{\sqrt{3} \sqrt{\sin{\left(x \right)}} \cos{\left(x \right)}}{2 \sin{\left(x \right)}} - \sin{\left(x \right)}$$

Simplify

The second derivative [src]

 /  ___   ________     ___    2            \
 |\/ 3 *\/ sin(x)    \/ 3 *cos (x)         |
-|---------------- + ------------- + cos(x)|
 |       2                 3/2             |
 \                    4*sin   (x)          /

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

Simplify

The third derivative [src]

  ___              ___    3            
\/ 3 *cos(x)   3*\/ 3 *cos (x)         
------------ + --------------- + sin(x)
    ________          5/2              
4*\/ sin(x)      8*sin   (x)

$$\sin{\left(x \right)} + \frac{\sqrt{3} \cos{\left(x \right)}}{4 \sqrt{\sin{\left(x \right)}}} + \frac{3 \sqrt{3} \cos^{3}{\left(x \right)}}{8 \sin^{\frac{5}{2}}{\left(x \right)}}$$

Simplify

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Derivative of √3sin(x)+cos(x)

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