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y=lncosx- one /x
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Similar expressions
y=lncosx+1/x

The derivative of the function/ y=lncosx-1/x

Derivative of y=lncosx-1/x

The solution

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              1
log(cos(x)) - -
              x

$$\log{\left(\cos{\left(x \right)} \right)} - \frac{1}{x}$$

log(cos(x)) - 1/x

Detail solution

Differentiate $\log{\left(\cos{\left(x \right)} \right)} - \frac{1}{x}$ term by term:
1. Let $u = \cos{\left(x \right)}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
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:
  
  $- \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
4. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $\frac{1}{x}$ goes to $- \frac{1}{x^{2}}$
  So, the result is: $\frac{1}{x^{2}}$
The result is: $- \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{1}{x^{2}}$
Now simplify:

$- \tan{\left(x \right)} + \frac{1}{x^{2}}$

The answer is:

$- \tan{\left(x \right)} + \frac{1}{x^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

1    sin(x)
-- - ------
 2   cos(x)
x

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

Simplify

The second derivative [src]

 /            2   \
 |    2    sin (x)|
-|1 + -- + -------|
 |     3      2   |
 \    x    cos (x)/

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

Simplify

The third derivative [src]

  /                 3   \
  |3    sin(x)   sin (x)|
2*|-- - ------ - -------|
  | 4   cos(x)      3   |
  \x             cos (x)/

$$2 \left(- \frac{\sin^{3}{\left(x \right)}}{\cos^{3}{\left(x \right)}} - \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{3}{x^{4}}\right)$$

Simplify

The graph

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Derivative of y=lncosx-1/x

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The solution