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The derivative of the function/ a^tant

Derivative of a^tant

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

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The solution

You have entered [src] 
 tan(t)
a
atan⁡(t)a^{\tan{\left(t \right)}} 
d / tan(t)\
--\a      /
dt
∂∂tatan⁡(t)\frac{\partial}{\partial t} a^{\tan{\left(t \right)}}
Transform
Detail solution

  1. Let u=tan⁡(t)u = \tan{\left(t \right)}.

  2. ∂∂uau=aulog⁡(a)\frac{\partial}{\partial u} a^{u} = a^{u} \log{\left(a \right)}

  3. Then, apply the chain rule. Multiply by ddttan⁡(t)\frac{d}{d t} \tan{\left(t \right)}:

    1. Rewrite the function to be differentiated:

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

    2. Apply the quotient rule, which is:

      ddtf(t)g(t)=−f(t)ddtg(t)+g(t)ddtf(t)g2(t)\frac{d}{d t} \frac{f{\left(t \right)}}{g{\left(t \right)}} = \frac{- f{\left(t \right)} \frac{d}{d t} g{\left(t \right)} + g{\left(t \right)} \frac{d}{d t} f{\left(t \right)}}{g^{2}{\left(t \right)}}

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

      To find ddtf(t)\frac{d}{d t} f{\left(t \right)}:

      1. The derivative of sine is cosine:

        ddtsin⁡(t)=cos⁡(t)\frac{d}{d t} \sin{\left(t \right)} = \cos{\left(t \right)}

      To find ddtg(t)\frac{d}{d t} g{\left(t \right)}:

      1. The derivative of cosine is negative sine:

        ddtcos⁡(t)=−sin⁡(t)\frac{d}{d t} \cos{\left(t \right)} = - \sin{\left(t \right)}

      Now plug in to the quotient rule:

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

    The result of the chain rule is:

    atan⁡(t)(sin⁡2(t)+cos⁡2(t))log⁡(a)cos⁡2(t)\frac{a^{\tan{\left(t \right)}} \left(\sin^{2}{\left(t \right)} + \cos^{2}{\left(t \right)}\right) \log{\left(a \right)}}{\cos^{2}{\left(t \right)}}

  4. Now simplify:

    atan⁡(t)log⁡(a)cos⁡2(t)\frac{a^{\tan{\left(t \right)}} \log{\left(a \right)}}{\cos^{2}{\left(t \right)}}

The answer is:

atan⁡(t)log⁡(a)cos⁡2(t)\frac{a^{\tan{\left(t \right)}} \log{\left(a \right)}}{\cos^{2}{\left(t \right)}}

The first derivative [src] 
 tan(t) /       2   \       
a      *\1 + tan (t)/*log(a)
atan⁡(t)(tan⁡2(t)+1)log⁡(a)a^{\tan{\left(t \right)}} \left(\tan^{2}{\left(t \right)} + 1\right) \log{\left(a \right)}
Simplify
The second derivative [src] 
 tan(t) /       2   \ /           /       2   \       \       
a      *\1 + tan (t)/*\2*tan(t) + \1 + tan (t)/*log(a)/*log(a)
atan⁡(t)((tan⁡2(t)+1)log⁡(a)+2tan⁡(t))(tan⁡2(t)+1)log⁡(a)a^{\tan{\left(t \right)}} \left(\left(\tan^{2}{\left(t \right)} + 1\right) \log{\left(a \right)} + 2 \tan{\left(t \right)}\right) \left(\tan^{2}{\left(t \right)} + 1\right) \log{\left(a \right)}
Simplify
The third derivative [src] 
                      /                             2                                        \       
 tan(t) /       2   \ |         2      /       2   \     2        /       2   \              |       
a      *\1 + tan (t)/*\2 + 6*tan (t) + \1 + tan (t)/ *log (a) + 6*\1 + tan (t)/*log(a)*tan(t)/*log(a)
atan⁡(t)(tan⁡2(t)+1)((tan⁡2(t)+1)2log⁡(a)2+6(tan⁡2(t)+1)log⁡(a)tan⁡(t)+6tan⁡2(t)+2)log⁡(a)a^{\tan{\left(t \right)}} \left(\tan^{2}{\left(t \right)} + 1\right) \left(\left(\tan^{2}{\left(t \right)} + 1\right)^{2} \log{\left(a \right)}^{2} + 6 \left(\tan^{2}{\left(t \right)} + 1\right) \log{\left(a \right)} \tan{\left(t \right)} + 6 \tan^{2}{\left(t \right)} + 2\right) \log{\left(a \right)}
Simplify
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