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e^(4cos(x-1))

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

Function f() - derivative -N order at the point
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The solution

You have entered [src]
 4*cos(x - 1)
e            
e4cos(x1)e^{4 \cos{\left(x - 1 \right)}}
d / 4*cos(x - 1)\
--\e            /
dx               
ddxe4cos(x1)\frac{d}{d x} e^{4 \cos{\left(x - 1 \right)}}
Detail solution
  1. Let u=4cos(x1)u = 4 \cos{\left(x - 1 \right)}.

  2. The derivative of eue^{u} is itself.

  3. Then, apply the chain rule. Multiply by ddx4cos(x1)\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=x1u = x - 1.

      2. The derivative of cosine is negative sine:

        dducos(u)=sin(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

      3. Then, apply the chain rule. Multiply by ddx(x1)\frac{d}{d x} \left(x - 1\right):

        1. Differentiate x1x - 1 term by term:

          1. Apply the power rule: xx goes to 11

          2. The derivative of the constant (1)1\left(-1\right) 1 is zero.

          The result is: 11

        The result of the chain rule is:

        sin(x1)- \sin{\left(x - 1 \right)}

      So, the result is: 4sin(x1)- 4 \sin{\left(x - 1 \right)}

    The result of the chain rule is:

    4e4cos(x1)sin(x1)- 4 e^{4 \cos{\left(x - 1 \right)}} \sin{\left(x - 1 \right)}

  4. Now simplify:

    4e4cos(x1)sin(x1)- 4 e^{4 \cos{\left(x - 1 \right)}} \sin{\left(x - 1 \right)}


The answer is:

4e4cos(x1)sin(x1)- 4 e^{4 \cos{\left(x - 1 \right)}} \sin{\left(x - 1 \right)}

The graph
02468-8-6-4-2-1010-100100
The first derivative [src]
    4*cos(x - 1)           
-4*e            *sin(x - 1)
4e4cos(x1)sin(x1)- 4 e^{4 \cos{\left(x - 1 \right)}} \sin{\left(x - 1 \right)}
The second derivative [src]
  /                    2        \  4*cos(-1 + x)
4*\-cos(-1 + x) + 4*sin (-1 + x)/*e             
4(4sin2(x1)cos(x1))e4cos(x1)4 \cdot \left(4 \sin^{2}{\left(x - 1 \right)} - \cos{\left(x - 1 \right)}\right) e^{4 \cos{\left(x - 1 \right)}}
The third derivative [src]
  /          2                         \  4*cos(-1 + x)            
4*\1 - 16*sin (-1 + x) + 12*cos(-1 + x)/*e             *sin(-1 + x)
4(16sin2(x1)+12cos(x1)+1)e4cos(x1)sin(x1)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)}
The graph
Derivative of e^(4cos(x-1))