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y=2cosx-3tgx
The derivative of the function/ y=2cosx-3tgx

Derivative of y=2cosx-3tgx

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You have entered [src] 
2*cos(x) - 3*tan(x)
2cos⁡(x)−3tan⁡(x)2 \cos{\left(x \right)} - 3 \tan{\left(x \right)} 
2*cos(x) - 3*tan(x)
Detail solution

  1. Differentiate 2cos⁡(x)−3tan⁡(x)2 \cos{\left(x \right)} - 3 \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:

        ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

      So, the result is: −2sin⁡(x)- 2 \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⁡(x)=sin⁡(x)cos⁡(x)\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}

      2. Apply the quotient rule, which is:

        ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=sin⁡(x)f{\left(x \right)} = \sin{\left(x \right)} and g(x)=cos⁡(x)g{\left(x \right)} = \cos{\left(x \right)}.

        To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

        1. The derivative of sine is cosine:

          ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

        To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

        1. The derivative of cosine is negative sine:

          ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

        Now plug in to the quotient rule:

        sin⁡2(x)+cos⁡2(x)cos⁡2(x)\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

      So, the result is: −3(sin⁡2(x)+cos⁡2(x))cos⁡2(x)- \frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}}

    The result is: −3(sin⁡2(x)+cos⁡2(x))cos⁡2(x)−2sin⁡(x)- \frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} - 2 \sin{\left(x \right)}

  2. Now simplify:

    2sin⁡3(x)−2sin⁡(x)−3cos⁡2(x)\frac{2 \sin^{3}{\left(x \right)} - 2 \sin{\left(x \right)} - 3}{\cos^{2}{\left(x \right)}}

The answer is:

2sin⁡3(x)−2sin⁡(x)−3cos⁡2(x)\frac{2 \sin^{3}{\left(x \right)} - 2 \sin{\left(x \right)} - 3}{\cos^{2}{\left(x \right)}}

The graph
02468-8-6-4-2-1010-25002500
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Plot the graph f'(x)
The first derivative [src] 
          2              
-3 - 3*tan (x) - 2*sin(x)
−2sin⁡(x)−3tan⁡2(x)−3- 2 \sin{\left(x \right)} - 3 \tan^{2}{\left(x \right)} - 3
Simplify
The second derivative [src] 
   /  /       2   \                \
-2*\3*\1 + tan (x)/*tan(x) + cos(x)/
−2(3(tan⁡2(x)+1)tan⁡(x)+cos⁡(x))- 2 \left(3 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \cos{\left(x \right)}\right)
Simplify
The third derivative [src] 
  /                 2                                   \
  |    /       2   \         2    /       2   \         |
2*\- 3*\1 + tan (x)/  - 6*tan (x)*\1 + tan (x)/ + sin(x)/
2(−3(tan⁡2(x)+1)2−6(tan⁡2(x)+1)tan⁡2(x)+sin⁡(x))2 \left(- 3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} - 6 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + \sin{\left(x \right)}\right)
Simplify
The graph
Derivative of y=2cosx-3tgx
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