Derivative of a^tant

The solution

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 tan(t)
a

$$a^{\tan{\left(t \right)}}$$

d / tan(t)\
--\a      /
dt

$$\frac{\partial}{\partial t} a^{\tan{\left(t \right)}}$$

Transform

Detail solution

Let $u = \tan{\left(t \right)}$ .
$\frac{\partial}{\partial u} a^{u} = a^{u} \log{\left(a \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d t} \tan{\left(t \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(t \right)} = \frac{\sin{\left(t \right)}}{\cos{\left(t \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d t} \frac{f{\left(t \right)}}{g{\left(t \right)}} = \frac{- f{\left(t \right)} \frac{d}{d t} g{\left(t \right)} + g{\left(t \right)} \frac{d}{d t} f{\left(t \right)}}{g^{2}{\left(t \right)}}$
  
  $f{\left(t \right)} = \sin{\left(t \right)}$ and $g{\left(t \right)} = \cos{\left(t \right)}$ .
  
  To find $\frac{d}{d t} f{\left(t \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d t} \sin{\left(t \right)} = \cos{\left(t \right)}$
  To find $\frac{d}{d t} g{\left(t \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d t} \cos{\left(t \right)} = - \sin{\left(t \right)}$
  Now plug in to the quotient rule:
  
  $\frac{\sin^{2}{\left(t \right)} + \cos^{2}{\left(t \right)}}{\cos^{2}{\left(t \right)}}$
The result of the chain rule is:

$\frac{a^{\tan{\left(t \right)}} \left(\sin^{2}{\left(t \right)} + \cos^{2}{\left(t \right)}\right) \log{\left(a \right)}}{\cos^{2}{\left(t \right)}}$
Now simplify:

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

The answer is:

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

The first derivative [src]

 tan(t) /       2   \       
a      *\1 + tan (t)/*log(a)

$$a^{\tan{\left(t \right)}} \left(\tan^{2}{\left(t \right)} + 1\right) \log{\left(a \right)}$$

Simplify

The second derivative [src]

 tan(t) /       2   \ /           /       2   \       \       
a      *\1 + tan (t)/*\2*tan(t) + \1 + tan (t)/*log(a)/*log(a)

$$a^{\tan{\left(t \right)}} \left(\left(\tan^{2}{\left(t \right)} + 1\right) \log{\left(a \right)} + 2 \tan{\left(t \right)}\right) \left(\tan^{2}{\left(t \right)} + 1\right) \log{\left(a \right)}$$

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)

$$a^{\tan{\left(t \right)}} \left(\tan^{2}{\left(t \right)} + 1\right) \left(\left(\tan^{2}{\left(t \right)} + 1\right)^{2} \log{\left(a \right)}^{2} + 6 \left(\tan^{2}{\left(t \right)} + 1\right) \log{\left(a \right)} \tan{\left(t \right)} + 6 \tan^{2}{\left(t \right)} + 2\right) \log{\left(a \right)}$$

Simplify

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Derivative of a^tant

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