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Derivative of f(x)=4cos(5x-1)

Function f() - derivative -N order at the point
v

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The solution

You have entered [src]
4*cos(5*x - 1)
4cos(5x1)4 \cos{\left(5 x - 1 \right)}
4*cos(5*x - 1)
Detail solution
  1. The derivative of a constant times a function is the constant times the derivative of the function.

    1. Let u=5x1u = 5 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(5x1)\frac{d}{d x} \left(5 x - 1\right):

      1. Differentiate 5x15 x - 1 term by term:

        1. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: xx goes to 11

          So, the result is: 55

        2. The derivative of the constant 1-1 is zero.

        The result is: 55

      The result of the chain rule is:

      5sin(5x1)- 5 \sin{\left(5 x - 1 \right)}

    So, the result is: 20sin(5x1)- 20 \sin{\left(5 x - 1 \right)}

  2. Now simplify:

    20sin(5x1)- 20 \sin{\left(5 x - 1 \right)}


The answer is:

20sin(5x1)- 20 \sin{\left(5 x - 1 \right)}

The graph
02468-8-6-4-2-1010-5050
The first derivative [src]
-20*sin(5*x - 1)
20sin(5x1)- 20 \sin{\left(5 x - 1 \right)}
The second derivative [src]
-100*cos(-1 + 5*x)
100cos(5x1)- 100 \cos{\left(5 x - 1 \right)}
The third derivative [src]
500*sin(-1 + 5*x)
500sin(5x1)500 \sin{\left(5 x - 1 \right)}