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Identical expressions
cos(lg(4x- five /2x+ one))
co sinus of e of (lg(4x minus 5 divide by 2x plus 1))
co sinus of e of (lg(4x minus five divide by 2x plus one))
coslg4x-5/2x+1
cos(lg(4x-5 divide by 2x+1))
Similar expressions
cos(lg(4x+5/2x+1))
cos(lg(4x-5/2x-1))

The derivative of the function/ cos(lg(4x-5/2x+1))

Derivative of cos(lg(4x-5/2x+1))

The solution

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   /   /      5*x    \\
cos|log|4*x - --- + 1||
   \   \       2     //

$$\cos{\left(\log{\left(\left(- \frac{5 x}{2} + 4 x\right) + 1 \right)} \right)}$$

cos(log(4*x - 5*x/2 + 1))

Detail solution

Let $u = \log{\left(\left(- \frac{5 x}{2} + 4 x\right) + 1 \right)}$ .
The derivative of cosine is negative sine:

$\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(\left(- \frac{5 x}{2} + 4 x\right) + 1 \right)}$ :
1. Let $u = \left(- \frac{5 x}{2} + 4 x\right) + 1$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\left(- \frac{5 x}{2} + 4 x\right) + 1\right)$ :
  1. Differentiate $\left(- \frac{5 x}{2} + 4 x\right) + 1$ term by term:
    1. Differentiate $- \frac{5 x}{2} + 4 x$ term by term:
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $4$
      2. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $- \frac{5}{2}$
      The result is: $\frac{3}{2}$
    2. The derivative of the constant $1$ is zero.
    The result is: $\frac{3}{2}$
  The result of the chain rule is:
  
  $\frac{3}{2 \left(\left(- \frac{5 x}{2} + 4 x\right) + 1\right)}$
The result of the chain rule is:

$- \frac{3 \sin{\left(\log{\left(\left(- \frac{5 x}{2} + 4 x\right) + 1 \right)} \right)}}{2 \left(\left(- \frac{5 x}{2} + 4 x\right) + 1\right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      /   /      5*x    \\
-3*sin|log|4*x - --- + 1||
      \   \       2     //
--------------------------
      /      5*x    \     
    2*|4*x - --- + 1|     
      \       2     /

$$- \frac{3 \sin{\left(\log{\left(\left(- \frac{5 x}{2} + 4 x\right) + 1 \right)} \right)}}{2 \left(\left(- \frac{5 x}{2} + 4 x\right) + 1\right)}$$

Simplify

The second derivative [src]

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

$$\frac{9 \left(\sin{\left(\log{\left(\frac{3 x}{2} + 1 \right)} \right)} - \cos{\left(\log{\left(\frac{3 x}{2} + 1 \right)} \right)}\right)}{\left(3 x + 2\right)^{2}}$$

Simplify

The third derivative [src]

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

$$\frac{27 \left(- \sin{\left(\log{\left(\frac{3 x}{2} + 1 \right)} \right)} + 3 \cos{\left(\log{\left(\frac{3 x}{2} + 1 \right)} \right)}\right)}{\left(3 x + 2\right)^{3}}$$

Simplify

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Derivative of cos(lg(4x-5/2x+1))

The graph:

Piecewise:

The solution