Mister Exam

Derivative of tan(5x)+sin(2x)+sec(x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
tan(5*x) + sin(2*x) + sec(x)
$$\left(\sin{\left(2 x \right)} + \tan{\left(5 x \right)}\right) + \sec{\left(x \right)}$$
tan(5*x) + sin(2*x) + sec(x)
Detail solution
  1. Differentiate term by term:

    1. Differentiate term by term:

      1. Rewrite the function to be differentiated:

      2. Apply the quotient rule, which is:

        and .

        To find :

        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:

        To find :

        1. Let .

        2. The derivative of cosine is negative sine:

        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:

        Now plug in to the quotient rule:

      3. Let .

      4. The derivative of sine is cosine:

      5. 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:

      The result is:

    2. Rewrite the function to be differentiated:

    3. Let .

    4. Apply the power rule: goes to

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

      1. The derivative of cosine is negative sine:

      The result of the chain rule is:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
                      2                     
5 + 2*cos(2*x) + 5*tan (5*x) + sec(x)*tan(x)
$$2 \cos{\left(2 x \right)} + \tan{\left(x \right)} \sec{\left(x \right)} + 5 \tan^{2}{\left(5 x \right)} + 5$$
The second derivative [src]
                 2             /       2   \             /       2     \         
-4*sin(2*x) + tan (x)*sec(x) + \1 + tan (x)/*sec(x) + 50*\1 + tan (5*x)/*tan(5*x)
$$\left(\tan^{2}{\left(x \right)} + 1\right) \sec{\left(x \right)} + 50 \left(\tan^{2}{\left(5 x \right)} + 1\right) \tan{\left(5 x \right)} - 4 \sin{\left(2 x \right)} + \tan^{2}{\left(x \right)} \sec{\left(x \right)}$$
The third derivative [src]
                                 2                                                                                 
                  /       2     \       3                    2      /       2     \     /       2   \              
-8*cos(2*x) + 250*\1 + tan (5*x)/  + tan (x)*sec(x) + 500*tan (5*x)*\1 + tan (5*x)/ + 5*\1 + tan (x)/*sec(x)*tan(x)
$$5 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \sec{\left(x \right)} + 250 \left(\tan^{2}{\left(5 x \right)} + 1\right)^{2} + 500 \left(\tan^{2}{\left(5 x \right)} + 1\right) \tan^{2}{\left(5 x \right)} - 8 \cos{\left(2 x \right)} + \tan^{3}{\left(x \right)} \sec{\left(x \right)}$$