Mister Exam

Derivative of tanxsecx

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
tan(x)*sec(x)
$$\tan{\left(x \right)} \sec{\left(x \right)}$$
tan(x)*sec(x)
Detail solution
  1. Apply the product rule:

    ; to find :

    1. Rewrite the function to be differentiated:

    2. Apply the quotient rule, which is:

      and .

      To find :

      1. The derivative of sine is cosine:

      To find :

      1. The derivative of cosine is negative sine:

      Now plug in to the quotient rule:

    ; to find :

    1. Rewrite the function to be differentiated:

    2. Let .

    3. Apply the power rule: goes to

    4. 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             /       2   \       
tan (x)*sec(x) + \1 + tan (x)/*sec(x)
$$\left(\tan^{2}{\left(x \right)} + 1\right) \sec{\left(x \right)} + \tan^{2}{\left(x \right)} \sec{\left(x \right)}$$
The second derivative [src]
/         2   \              
\5 + 6*tan (x)/*sec(x)*tan(x)
$$\left(6 \tan^{2}{\left(x \right)} + 5\right) \tan{\left(x \right)} \sec{\left(x \right)}$$
The third derivative [src]
/   2    /         2   \     /       2   \ /         2   \     /       2   \ /         2   \        2    /       2   \\       
\tan (x)*\5 + 6*tan (x)/ + 2*\1 + tan (x)/*\1 + 3*tan (x)/ + 3*\1 + tan (x)/*\1 + 2*tan (x)/ + 6*tan (x)*\1 + tan (x)//*sec(x)
$$\left(3 \left(\tan^{2}{\left(x \right)} + 1\right) \left(2 \tan^{2}{\left(x \right)} + 1\right) + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) + 6 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + \left(6 \tan^{2}{\left(x \right)} + 5\right) \tan^{2}{\left(x \right)}\right) \sec{\left(x \right)}$$
The graph
Derivative of tanxsecx