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The derivative of the function/ sin(5*x+3*x^3)

Derivative of sin(5*x+3*x^3)

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   /         3\
sin\5*x + 3*x /
$$\sin{\left(3 x^{3} + 5 x \right)}$$ 
sin(5*x + 3*x^3)
Detail solution

  1. Let .

  2. The derivative of sine is cosine:

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

    1. Differentiate 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: goes to

        So, the result is:

      2. 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 is:

    The result of the chain rule is:

  4. Now simplify:

The answer is:

The graph
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
/       2\    /         3\
\5 + 9*x /*cos\5*x + 3*x /
$$\left(9 x^{2} + 5\right) \cos{\left(3 x^{3} + 5 x \right)}$$
Simplify
The second derivative [src] 
            2                                           
  /       2\     /  /       2\\           /  /       2\\
- \5 + 9*x / *sin\x*\5 + 3*x // + 18*x*cos\x*\5 + 3*x //
$$18 x \cos{\left(x \left(3 x^{2} + 5\right) \right)} - \left(9 x^{2} + 5\right)^{2} \sin{\left(x \left(3 x^{2} + 5\right) \right)}$$
Simplify
The third derivative [src] 
                                 3                                                      
      /  /       2\\   /       2\     /  /       2\\        /       2\    /  /       2\\
18*cos\x*\5 + 3*x // - \5 + 9*x / *cos\x*\5 + 3*x // - 54*x*\5 + 9*x /*sin\x*\5 + 3*x //
$$- 54 x \left(9 x^{2} + 5\right) \sin{\left(x \left(3 x^{2} + 5\right) \right)} - \left(9 x^{2} + 5\right)^{3} \cos{\left(x \left(3 x^{2} + 5\right) \right)} + 18 \cos{\left(x \left(3 x^{2} + 5\right) \right)}$$
Simplify
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