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tanxsecx
The derivative of the function/ tanxsecx

Derivative of tanxsecx

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

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The solution

You have entered [src] 
tan(x)*sec(x)
tan⁡(x)sec⁡(x)\tan{\left(x \right)} \sec{\left(x \right)} 
tan(x)*sec(x)
Detail solution

  1. Apply the product rule:

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

    f(x)=tan⁡(x)f{\left(x \right)} = \tan{\left(x \right)}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    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:

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

    g(x)=sec⁡(x)g{\left(x \right)} = \sec{\left(x \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Rewrite the function to be differentiated:

      sec⁡(x)=1cos⁡(x)\sec{\left(x \right)} = \frac{1}{\cos{\left(x \right)}}

    2. Let u=cos⁡(x)u = \cos{\left(x \right)}.

    3. Apply the power rule: 1u\frac{1}{u} goes to −1u2- \frac{1}{u^{2}}

    4. Then, apply the chain rule. Multiply by ddxcos⁡(x)\frac{d}{d x} \cos{\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)}

      The result of the chain rule is:

      sin⁡(x)cos⁡2(x)\frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}

    The result is: (sin⁡2(x)+cos⁡2(x))sec⁡(x)cos⁡2(x)+sin⁡(x)tan⁡(x)cos⁡2(x)\frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sec{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin{\left(x \right)} \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}}

  2. Now simplify:

    −1+2cos⁡2(x)cos⁡(x)\frac{-1 + \frac{2}{\cos^{2}{\left(x \right)}}}{\cos{\left(x \right)}}

The answer is:

−1+2cos⁡2(x)cos⁡(x)\frac{-1 + \frac{2}{\cos^{2}{\left(x \right)}}}{\cos{\left(x \right)}}

The graph
02468-8-6-4-2-1010-5000050000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   2             /       2   \       
tan (x)*sec(x) + \1 + tan (x)/*sec(x)
(tan⁡2(x)+1)sec⁡(x)+tan⁡2(x)sec⁡(x)\left(\tan^{2}{\left(x \right)} + 1\right) \sec{\left(x \right)} + \tan^{2}{\left(x \right)} \sec{\left(x \right)}
Simplify
The second derivative [src] 
/         2   \              
\5 + 6*tan (x)/*sec(x)*tan(x)
(6tan⁡2(x)+5)tan⁡(x)sec⁡(x)\left(6 \tan^{2}{\left(x \right)} + 5\right) \tan{\left(x \right)} \sec{\left(x \right)}
Simplify
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)
(3(tan⁡2(x)+1)(2tan⁡2(x)+1)+2(tan⁡2(x)+1)(3tan⁡2(x)+1)+6(tan⁡2(x)+1)tan⁡2(x)+(6tan⁡2(x)+5)tan⁡2(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)}
Simplify
The graph
Derivative of tanxsecx
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