Derivative of y=4cosx-5tgx+3

The solution

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4*cos(x) - 5*tan(x) + 3

$$\left(4 \cos{\left(x \right)} - 5 \tan{\left(x \right)}\right) + 3$$

4*cos(x) - 5*tan(x) + 3

Detail solution

Differentiate $\left(4 \cos{\left(x \right)} - 5 \tan{\left(x \right)}\right) + 3$ term by term:
1. Differentiate $4 \cos{\left(x \right)} - 5 \tan{\left(x \right)}$ term by term:
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    So, the result is: $- 4 \sin{\left(x \right)}$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Rewrite the function to be differentiated:
      
      $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
    2. Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = \cos{\left(x \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      1. The derivative of sine is cosine:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      1. The derivative of cosine is negative sine:
        
        $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      Now plug in to the quotient rule:
      
      $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
    So, the result is: $- \frac{5 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}}$
  The result is: $- \frac{5 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} - 4 \sin{\left(x \right)}$
2. The derivative of the constant $3$ is zero.
The result is: $- \frac{5 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} - 4 \sin{\left(x \right)}$
Now simplify:

$- \frac{\sin{\left(x \right)} + \sin{\left(3 x \right)} + 5}{\cos^{2}{\left(x \right)}}$

The answer is:

$- \frac{\sin{\left(x \right)} + \sin{\left(3 x \right)} + 5}{\cos^{2}{\left(x \right)}}$

The graph

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The first derivative [src]

          2              
-5 - 5*tan (x) - 4*sin(x)

$$- 4 \sin{\left(x \right)} - 5 \tan^{2}{\left(x \right)} - 5$$

Simplify

The second derivative [src]

   /             /       2   \       \
-2*\2*cos(x) + 5*\1 + tan (x)/*tan(x)/

$$- 2 \left(5 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 2 \cos{\left(x \right)}\right)$$

Simplify

The third derivative [src]

  /                 2                                      \
  |    /       2   \                     2    /       2   \|
2*\- 5*\1 + tan (x)/  + 2*sin(x) - 10*tan (x)*\1 + tan (x)//

$$2 \left(- 5 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} - 10 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 2 \sin{\left(x \right)}\right)$$

Simplify

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Derivative of y=4cosx-5tgx+3

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