Derivative of 5sinx-4tanx

The solution

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

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

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

Detail solution

Differentiate $5 \sin{\left(x \right)} - 4 \tan{\left(x \right)}$ 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 sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  So, the result is: $5 \cos{\left(x \right)}$
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: $- \frac{4 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}}$
The result is: $- \frac{4 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + 5 \cos{\left(x \right)}$
Now simplify:

$5 \cos{\left(x \right)} - \frac{4}{\cos^{2}{\left(x \right)}}$

The answer is:

$5 \cos{\left(x \right)} - \frac{4}{\cos^{2}{\left(x \right)}}$

The graph

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The first derivative [src]

          2              
-4 - 4*tan (x) + 5*cos(x)

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

Simplify

The second derivative [src]

 /             /       2   \       \
-\5*sin(x) + 8*\1 + tan (x)/*tan(x)/

$$- (8 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 5 \sin{\left(x \right)})$$

Simplify

The third derivative [src]

 /                          2                           \
 |             /       2   \          2    /       2   \|
-\5*cos(x) + 8*\1 + tan (x)/  + 16*tan (x)*\1 + tan (x)//

$$- (8 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 16 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 5 \cos{\left(x \right)})$$

Simplify

The graph

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Derivative of 5sinx-4tanx

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