Derivative of y=e^cosx+lnx

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 cos(x)         
E       + log(x)

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

E^cos(x) + log(x)

Detail solution

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

1    cos(x)       
- - e      *sin(x)
x

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

Simplify

The second derivative [src]

  1       2     cos(x)           cos(x)
- -- + sin (x)*e       - cos(x)*e      
   2                                   
  x

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

Simplify

The third derivative [src]

2     cos(x)             3     cos(x)             cos(x)       
-- + e      *sin(x) - sin (x)*e       + 3*cos(x)*e      *sin(x)
 3                                                             
x

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

Simplify

The graph

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Derivative of y=e^cosx+lnx

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