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The derivative of the function/ 2*t*sin(4*x)+1/x

Derivative of 2*t*sin(4*x)+1/x

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
               1
2*t*sin(4*x) + -
               x
$$2 t \sin{\left(4 x \right)} + \frac{1}{x}$$ 
(2*t)*sin(4*x) + 1/x
Detail solution

  1. Differentiate term by term:

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

      1. Let .

      2. The derivative of sine is cosine:

      3. Then, apply the chain rule. Multiply by :

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

          1. Apply the power rule: goes to

          So, the result is:

        The result of the chain rule is:

      So, the result is:

    2. Apply the power rule: goes to

    The result is:

The answer is:

The first derivative [src] 
  1                
- -- + 8*t*cos(4*x)
   2               
  x
$$8 t \cos{\left(4 x \right)} - \frac{1}{x^{2}}$$
Simplify
The second derivative [src] 
  /1                 \
2*|-- - 16*t*sin(4*x)|
  | 3                |
  \x                 /
$$2 \left(- 16 t \sin{\left(4 x \right)} + \frac{1}{x^{3}}\right)$$
Simplify
The third derivative [src] 
   /3                 \
-2*|-- + 64*t*cos(4*x)|
   | 4                |
   \x                 /
$$- 2 \left(64 t \cos{\left(4 x \right)} + \frac{3}{x^{4}}\right)$$
Simplify
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