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y=sqrt(x)sin(x+ six)
y equally square root of (x) sinus of (x plus 6)
y equally square root of (x) sinus of (x plus six)
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y=sqrtxsinx+6
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y=sqrt(x)sin(x-6)

The derivative of the function/ y=sqrt(x)sin(x+6)

Derivative of y=sqrt(x)sin(x+6)

The solution

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  ___           
\/ x *sin(x + 6)

$$\sqrt{x} \sin{\left(x + 6 \right)}$$

sqrt(x)*sin(x + 6)

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = \sqrt{x}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
$g{\left(x \right)} = \sin{\left(x + 6 \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x + 6$ .
2. The derivative of sine is cosine:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 6\right)$ :
  1. Differentiate $x + 6$ term by term:
    1. Apply the power rule: $x$ goes to $1$
    2. The derivative of the constant $6$ is zero.
    The result is: $1$
  The result of the chain rule is:
  
  $\cos{\left(x + 6 \right)}$
The result is: $\sqrt{x} \cos{\left(x + 6 \right)} + \frac{\sin{\left(x + 6 \right)}}{2 \sqrt{x}}$
Now simplify:

$\frac{x \cos{\left(x + 6 \right)} + \frac{\sin{\left(x + 6 \right)}}{2}}{\sqrt{x}}$

The answer is:

$\frac{x \cos{\left(x + 6 \right)} + \frac{\sin{\left(x + 6 \right)}}{2}}{\sqrt{x}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  ___              sin(x + 6)
\/ x *cos(x + 6) + ----------
                        ___  
                    2*\/ x

$$\sqrt{x} \cos{\left(x + 6 \right)} + \frac{\sin{\left(x + 6 \right)}}{2 \sqrt{x}}$$

Simplify

The second derivative [src]

cos(6 + x)     ___              sin(6 + x)
---------- - \/ x *sin(6 + x) - ----------
    ___                              3/2  
  \/ x                            4*x

$$- \sqrt{x} \sin{\left(x + 6 \right)} + \frac{\cos{\left(x + 6 \right)}}{\sqrt{x}} - \frac{\sin{\left(x + 6 \right)}}{4 x^{\frac{3}{2}}}$$

Simplify

The third derivative [src]

    ___              3*sin(6 + x)   3*cos(6 + x)   3*sin(6 + x)
- \/ x *cos(6 + x) - ------------ - ------------ + ------------
                           ___            3/2            5/2   
                       2*\/ x          4*x            8*x

$$- \sqrt{x} \cos{\left(x + 6 \right)} - \frac{3 \sin{\left(x + 6 \right)}}{2 \sqrt{x}} - \frac{3 \cos{\left(x + 6 \right)}}{4 x^{\frac{3}{2}}} + \frac{3 \sin{\left(x + 6 \right)}}{8 x^{\frac{5}{2}}}$$

Simplify

The graph

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Derivative of y=sqrt(x)sin(x+6)

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The solution