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sin(x)^ five ^(one / two)
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Similar expressions
sinx^5^(1/2)

The derivative of the function/ sin(x)^5^(1/2)

Derivative of sin(x)^5^(1/2)

The solution

You have entered [src]

          ___
        \/ 5 
(sin(x))

$$\sin^{\sqrt{5}}{\left(x \right)}$$

  /          ___\
d |        \/ 5 |
--\(sin(x))     /
dx

$$\frac{d}{d x} \sin^{\sqrt{5}}{\left(x \right)}$$

Transform

Detail solution

Let $u = \sin{\left(x \right)}$ .
Apply the power rule: $u^{\sqrt{5}}$ goes to $\frac{\sqrt{5} u^{\sqrt{5}}}{u}$
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:

$\frac{\sqrt{5} \sin^{\sqrt{5}}{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}}$
Now simplify:

$\sqrt{5} \sin^{-1 + \sqrt{5}}{\left(x \right)} \cos{\left(x \right)}$

The answer is:

$\sqrt{5} \sin^{-1 + \sqrt{5}}{\left(x \right)} \cos{\left(x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                ___       
  ___         \/ 5        
\/ 5 *(sin(x))     *cos(x)
--------------------------
          sin(x)

$$\frac{\sqrt{5} \sin^{\sqrt{5}}{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}}$$

Simplify

The second derivative [src]

          ___ /               2        ___    2   \
        \/ 5  |    ___   5*cos (x)   \/ 5 *cos (x)|
(sin(x))     *|- \/ 5  + --------- - -------------|
              |              2             2      |
              \           sin (x)       sin (x)   /

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

Simplify

The third derivative [src]

          ___ /                      2          ___    2   \       
        \/ 5  |          ___   15*cos (x)   7*\/ 5 *cos (x)|       
(sin(x))     *|-15 + 2*\/ 5  - ---------- + ---------------|*cos(x)
              |                    2               2       |       
              \                 sin (x)         sin (x)    /       
-------------------------------------------------------------------
                               sin(x)

$$\frac{\left(-15 + 2 \sqrt{5} - \frac{15 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{7 \sqrt{5} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \sin^{\sqrt{5}}{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}}$$

Simplify

The graph

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Derivative of sin(x)^5^(1/2)

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The solution