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y=e^(cost+lnt)
The derivative of the function/ y=e^(cost+lnt)

Derivative of y=e^(cost+lnt)

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
 cos(t) + log(t)
e
elog⁡(t)+cos⁡(t)e^{\log{\left(t \right)} + \cos{\left(t \right)}} 
d / cos(t) + log(t)\
--\e               /
dt
ddtelog⁡(t)+cos⁡(t)\frac{d}{d t} e^{\log{\left(t \right)} + \cos{\left(t \right)}}
Transform
Detail solution

  1. Let u=log⁡(t)+cos⁡(t)u = \log{\left(t \right)} + \cos{\left(t \right)}.

  2. The derivative of eue^{u} is itself.

  3. Then, apply the chain rule. Multiply by ddt(log⁡(t)+cos⁡(t))\frac{d}{d t} \left(\log{\left(t \right)} + \cos{\left(t \right)}\right):

    1. Differentiate log⁡(t)+cos⁡(t)\log{\left(t \right)} + \cos{\left(t \right)} term by term:

      1. The derivative of cosine is negative sine:

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

      2. The derivative of log⁡(t)\log{\left(t \right)} is 1t\frac{1}{t}.

      The result is: −sin⁡(t)+1t- \sin{\left(t \right)} + \frac{1}{t}

    The result of the chain rule is:

    t(−sin⁡(t)+1t)ecos⁡(t)t \left(- \sin{\left(t \right)} + \frac{1}{t}\right) e^{\cos{\left(t \right)}}

  4. Now simplify:

    (−tsin⁡(t)+1)ecos⁡(t)\left(- t \sin{\left(t \right)} + 1\right) e^{\cos{\left(t \right)}}

The answer is:

(−tsin⁡(t)+1)ecos⁡(t)\left(- t \sin{\left(t \right)} + 1\right) e^{\cos{\left(t \right)}}

The graph
02468-8-6-4-2-1010-5050
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   cos(t) /1         \
t*e      *|- - sin(t)|
          \t         /
tecos⁡(t)(−sin⁡(t)+1t)t e^{\cos{\left(t \right)}} \left(- \sin{\left(t \right)} + \frac{1}{t}\right)
Simplify
The second derivative [src] 
/                /  1         \     /1          \\  cos(t)
|(-1 + t*sin(t))*|- - + sin(t)| - t*|-- + cos(t)||*e      
|                \  t         /     | 2         ||        
\                                   \t          //
(−t(cos⁡(t)+1t2)+(tsin⁡(t)−1)(sin⁡(t)−1t))ecos⁡(t)\left(- t \left(\cos{\left(t \right)} + \frac{1}{t^{2}}\right) + \left(t \sin{\left(t \right)} - 1\right) \left(\sin{\left(t \right)} - \frac{1}{t}\right)\right) e^{\cos{\left(t \right)}}
Simplify
The third derivative [src] 
/  1               /2          \                   /1          \   /  1         \ /                           2   \     /1          \       \  cos(t)
|- -- - cos(t) + t*|-- + sin(t)| + (-1 + t*sin(t))*|-- + cos(t)| + |- - + sin(t)|*\2*sin(t) + t*cos(t) - t*sin (t)/ + t*|-- + cos(t)|*sin(t)|*e      
|   2              | 3         |                   | 2         |   \  t         /                                       | 2         |       |        
\  t               \t          /                   \t          /                                                        \t          /       /
(t(cos⁡(t)+1t2)sin⁡(t)+t(sin⁡(t)+2t3)+(tsin⁡(t)−1)(cos⁡(t)+1t2)+(sin⁡(t)−1t)(−tsin⁡2(t)+tcos⁡(t)+2sin⁡(t))−cos⁡(t)−1t2)ecos⁡(t)\left(t \left(\cos{\left(t \right)} + \frac{1}{t^{2}}\right) \sin{\left(t \right)} + t \left(\sin{\left(t \right)} + \frac{2}{t^{3}}\right) + \left(t \sin{\left(t \right)} - 1\right) \left(\cos{\left(t \right)} + \frac{1}{t^{2}}\right) + \left(\sin{\left(t \right)} - \frac{1}{t}\right) \left(- t \sin^{2}{\left(t \right)} + t \cos{\left(t \right)} + 2 \sin{\left(t \right)}\right) - \cos{\left(t \right)} - \frac{1}{t^{2}}\right) e^{\cos{\left(t \right)}}
Simplify
The graph
Derivative of y=e^(cost+lnt)
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