Mister Exam

Derivative of tan(x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
tan(x)
tan(x)\tan{\left(x \right)}
tan(x)
Detail solution
  1. Rewrite the function to be differentiated:

    tan(x)=sin(x)cos(x)\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}

  2. Apply the quotient rule, which is:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)g2(x)\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}

    f(x)=sin(x)f{\left(x \right)} = \sin{\left(x \right)} and g(x)=cos(x)g{\left(x \right)} = \cos{\left(x \right)}.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. The derivative of sine is cosine:

      ddxsin(x)=cos(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. The derivative of cosine is negative sine:

      ddxcos(x)=sin(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

    Now plug in to the quotient rule:

    sin2(x)+cos2(x)cos2(x)\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

  3. Now simplify:

    1cos2(x)\frac{1}{\cos^{2}{\left(x \right)}}


The answer is:

1cos2(x)\frac{1}{\cos^{2}{\left(x \right)}}

The graph
02468-8-6-4-2-1010-10001000
The first derivative [src]
       2   
1 + tan (x)
tan2(x)+1\tan^{2}{\left(x \right)} + 1
The second derivative [src]
  /       2   \       
2*\1 + tan (x)/*tan(x)
2(tan2(x)+1)tan(x)2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}
The third derivative [src]
  /       2   \ /         2   \
2*\1 + tan (x)/*\1 + 3*tan (x)/
2(tan2(x)+1)(3tan2(x)+1)2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right)
The graph
Derivative of tan(x)