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Derivative of y=e^x*sinx-3sqrt(x)

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 x              ___
e *sin(x) - 3*\/ x

$$e^{x} \sin{\left(x \right)} - 3 \sqrt{x}$$

d / x              ___\
--\e *sin(x) - 3*\/ x /
dx

$$\frac{d}{d x} \left(e^{x} \sin{\left(x \right)} - 3 \sqrt{x}\right)$$

Transform

Detail solution

Differentiate $- 3 \sqrt{x} + e^{x} \sin{\left(x \right)}$ term by term:
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = e^{x}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of $e^{x}$ is itself.
  $g{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result is: $e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
    So, the result is: $\frac{3}{2 \sqrt{x}}$
  So, the result is: $- \frac{3}{2 \sqrt{x}}$
The result is: $e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)} - \frac{3}{2 \sqrt{x}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  3                x
------ + 2*cos(x)*e 
   3/2              
4*x

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

Simplify

The third derivative [src]

    9         x                    x
- ------ - 2*e *sin(x) + 2*cos(x)*e 
     5/2                            
  8*x

$$- 2 e^{x} \sin{\left(x \right)} + 2 e^{x} \cos{\left(x \right)} - \frac{9}{8 x^{\frac{5}{2}}}$$

Simplify

The graph