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Derivative of sqrt(x)*(2*sin(x)+1)
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Identical expressions
sqrt(x)*(two *sin(x)+ one)
square root of (x) multiply by (2 multiply by sinus of (x) plus 1)
square root of (x) multiply by (two multiply by sinus of (x) plus one)
√(x)*(2*sin(x)+1)
sqrt(x)(2sin(x)+1)
sqrtx2sinx+1
Similar expressions
sqrt(x)*(2*sin(x)-1)
y=sqrt(x)(2sinx+1)
y=(sqrt(x))*(2sinx+1)
sqrt(x)*(2*sinx+1)

The derivative of the function/ sqrt(x)*(2*sin(x)+1)

Derivative of sqrt(x)(2sin(x)+1)

The solution

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  ___               
\/ x *(2*sin(x) + 1)

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

sqrt(x)*(2*sin(x) + 1)

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 $\frac{1}{2 \sqrt{x}}$
$g{\left(x \right)} = 2 \sin{\left(x \right)} + 1$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $2 \sin{\left(x \right)} + 1$ term by term:
  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: $2 \cos{\left(x \right)}$
  2. The derivative of the constant $1$ is zero.
  The result is: $2 \cos{\left(x \right)}$
The result is: $2 \sqrt{x} \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)} + 1}{2 \sqrt{x}}$
Now simplify:

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

The answer is:

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

The graph

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The first derivative [src]

2*sin(x) + 1       ___       
------------ + 2*\/ x *cos(x)
      ___                    
  2*\/ x

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  3*sin(x)       ___          3*cos(x)   3*(1 + 2*sin(x))
- -------- - 2*\/ x *cos(x) - -------- + ----------------
     ___                          3/2            5/2     
   \/ x                        2*x            8*x

$$- 2 \sqrt{x} \cos{\left(x \right)} - \frac{3 \sin{\left(x \right)}}{\sqrt{x}} - \frac{3 \cos{\left(x \right)}}{2 x^{\frac{3}{2}}} + \frac{3 \left(2 \sin{\left(x \right)} + 1\right)}{8 x^{\frac{5}{2}}}$$

Simplify

The graph

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Derivative of sqrt(x)(2sin(x)+1)

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Piecewise:

The solution

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Identical expressions

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Derivative of sqrt(x)*(2*sin(x)+1)

The graph:

Piecewise:

The solution

Derivative of sqrt(x)(2sin(x)+1)