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Derivative of:
Derivative of e^cos(x)
Derivative of ln(x+3)^3
Derivative of cos(x)-1
Derivative of cos(3*x)^(2)
Identical expressions
e^(4cos(x- one))
e to the power of (4 co sinus of e of (x minus 1))
e to the power of (4 co sinus of e of (x minus one))
e(4cos(x-1))
e4cosx-1
e^4cosx-1
Similar expressions
e^(4cos(x+1))

The derivative of the function/ e^(4cos(x-1))

Derivative of e^(4cos(x-1))

The solution

You have entered [src]

 4*cos(x - 1)
e

$$e^{4 \cos{\left(x - 1 \right)}}$$

d / 4*cos(x - 1)\
--\e            /
dx

$$\frac{d}{d x} e^{4 \cos{\left(x - 1 \right)}}$$

Transform

Detail solution

Let $u = 4 \cos{\left(x - 1 \right)}$ .
The derivative of $e^{u}$ is itself.
Then, apply the chain rule. Multiply by $\frac{d}{d x} 4 \cos{\left(x - 1 \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = x - 1$ .
  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 $\frac{d}{d x} \left(x - 1\right)$ :
    1. Differentiate $x - 1$ term by term:
      1. Apply the power rule: $x$ goes to $1$
      2. The derivative of the constant $\left(-1\right) 1$ is zero.
      The result is: $1$
    The result of the chain rule is:
    
    $- \sin{\left(x - 1 \right)}$
  So, the result is: $- 4 \sin{\left(x - 1 \right)}$
The result of the chain rule is:

$- 4 e^{4 \cos{\left(x - 1 \right)}} \sin{\left(x - 1 \right)}$
Now simplify:

$- 4 e^{4 \cos{\left(x - 1 \right)}} \sin{\left(x - 1 \right)}$

The answer is:

$- 4 e^{4 \cos{\left(x - 1 \right)}} \sin{\left(x - 1 \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    4*cos(x - 1)           
-4*e            *sin(x - 1)

$$- 4 e^{4 \cos{\left(x - 1 \right)}} \sin{\left(x - 1 \right)}$$

Simplify

The second derivative [src]

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

$$4 \cdot \left(4 \sin^{2}{\left(x - 1 \right)} - \cos{\left(x - 1 \right)}\right) e^{4 \cos{\left(x - 1 \right)}}$$

Simplify

The third derivative [src]

  /          2                         \  4*cos(-1 + x)            
4*\1 - 16*sin (-1 + x) + 12*cos(-1 + x)/*e             *sin(-1 + x)

$$4 \left(- 16 \sin^{2}{\left(x - 1 \right)} + 12 \cos{\left(x - 1 \right)} + 1\right) e^{4 \cos{\left(x - 1 \right)}} \sin{\left(x - 1 \right)}$$

Simplify

The graph

Derivative of e^(4cos(x-1))