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Derivative of:
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Identical expressions
e^sin(three *x)
e to the power of sinus of (3 multiply by x)
e to the power of sinus of (three multiply by x)
esin(3*x)
esin3*x
e^sin(3x)
esin(3x)
esin3x
e^sin3x
Similar expressions
y=x^2ln(e^sin(3x))

The derivative of the function/ e^sin(3*x)

Derivative of e^sin(3*x)

The solution

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 sin(3*x)
e

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

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

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

Transform

Detail solution

Let $u = \sin{\left(3 x \right)}$ .
The derivative of $e^{u}$ is itself.
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(3 x \right)}$ :
1. Let $u = 3 x$ .
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 $\frac{d}{d x} 3 x$ :
  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:
  
  $3 \cos{\left(3 x \right)}$
The result of the chain rule is:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Derivative of e^sin(3*x)