Derivative of y=e^(cost+lnt)

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 cos(t) + log(t)
e

$$e^{\log{\left(t \right)} + \cos{\left(t \right)}}$$

d / cos(t) + log(t)\
--\e               /
dt

$$\frac{d}{d t} e^{\log{\left(t \right)} + \cos{\left(t \right)}}$$

Transform

Detail solution

Let $u = \log{\left(t \right)} + \cos{\left(t \right)}$ .
The derivative of $e^{u}$ is itself.
Then, apply the chain rule. Multiply by $\frac{d}{d t} \left(\log{\left(t \right)} + \cos{\left(t \right)}\right)$ :
1. Differentiate $\log{\left(t \right)} + \cos{\left(t \right)}$ term by term:
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d t} \cos{\left(t \right)} = - \sin{\left(t \right)}$
  2. The derivative of $\log{\left(t \right)}$ is $\frac{1}{t}$ .
  The result is: $- \sin{\left(t \right)} + \frac{1}{t}$
The result of the chain rule is:

$t \left(- \sin{\left(t \right)} + \frac{1}{t}\right) e^{\cos{\left(t \right)}}$
Now simplify:

$\left(- t \sin{\left(t \right)} + 1\right) e^{\cos{\left(t \right)}}$

The answer is:

$\left(- t \sin{\left(t \right)} + 1\right) e^{\cos{\left(t \right)}}$

The graph

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The first derivative [src]

   cos(t) /1         \
t*e      *|- - sin(t)|
          \t         /

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

Simplify

The second derivative [src]

/                /  1         \     /1          \\  cos(t)
|(-1 + t*sin(t))*|- - + sin(t)| - t*|-- + cos(t)||*e      
|                \  t         /     | 2         ||        
\                                   \t          //

$$\left(- t \left(\cos{\left(t \right)} + \frac{1}{t^{2}}\right) + \left(t \sin{\left(t \right)} - 1\right) \left(\sin{\left(t \right)} - \frac{1}{t}\right)\right) e^{\cos{\left(t \right)}}$$

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          /       /

$$\left(t \left(\cos{\left(t \right)} + \frac{1}{t^{2}}\right) \sin{\left(t \right)} + t \left(\sin{\left(t \right)} + \frac{2}{t^{3}}\right) + \left(t \sin{\left(t \right)} - 1\right) \left(\cos{\left(t \right)} + \frac{1}{t^{2}}\right) + \left(\sin{\left(t \right)} - \frac{1}{t}\right) \left(- t \sin^{2}{\left(t \right)} + t \cos{\left(t \right)} + 2 \sin{\left(t \right)}\right) - \cos{\left(t \right)} - \frac{1}{t^{2}}\right) e^{\cos{\left(t \right)}}$$

Simplify

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Derivative of y=e^(cost+lnt)

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