Derivative of y=e^(sinx)cos3x

The solution

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 sin(x)         
E      *cos(3*x)

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

E^sin(x)*cos(3*x)

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     sin(x)                             sin(x)
- 3*e      *sin(3*x) + cos(x)*cos(3*x)*e

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

Simplify

The second derivative [src]

 /             /     2            \                             \  sin(x)
-\9*cos(3*x) + \- cos (x) + sin(x)/*cos(3*x) + 6*cos(x)*sin(3*x)/*e

$$- \left(\left(\sin{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos{\left(3 x \right)} + 6 \sin{\left(3 x \right)} \cos{\left(x \right)} + 9 \cos{\left(3 x \right)}\right) e^{\sin{\left(x \right)}}$$

Simplify

The third derivative [src]

/                                     /     2            \            /       2              \                \  sin(x)
\27*sin(3*x) - 27*cos(x)*cos(3*x) + 9*\- cos (x) + sin(x)/*sin(3*x) - \1 - cos (x) + 3*sin(x)/*cos(x)*cos(3*x)/*e

$$\left(9 \left(\sin{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \sin{\left(3 x \right)} - \left(3 \sin{\left(x \right)} - \cos^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} \cos{\left(3 x \right)} + 27 \sin{\left(3 x \right)} - 27 \cos{\left(x \right)} \cos{\left(3 x \right)}\right) e^{\sin{\left(x \right)}}$$

Simplify

The graph

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Derivative of y=e^(sinx)cos3x

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The solution