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The derivative of the function/ y=(1-tgx)*(lnx+2^x)

Derivative of y=(1-tgx)*(lnx+2^x)

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
             /          x\
(1 - tan(x))*\log(x) + 2 /
(1−tan⁡(x))(2x+log⁡(x))\left(1 - \tan{\left(x \right)}\right) \left(2^{x} + \log{\left(x \right)}\right) 
(1 - tan(x))*(log(x) + 2^x)
Detail solution

  1. Apply the product rule:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}

    f(x)=1−tan⁡(x)f{\left(x \right)} = 1 - \tan{\left(x \right)}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Differentiate 1−tan⁡(x)1 - \tan{\left(x \right)} term by term:

      1. The derivative of the constant 11 is zero.

      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: −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)}}

      The result is: −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)}}

    g(x)=2x+log⁡(x)g{\left(x \right)} = 2^{x} + \log{\left(x \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Differentiate 2x+log⁡(x)2^{x} + \log{\left(x \right)} term by term:

      1. The derivative of log⁡(x)\log{\left(x \right)} is 1x\frac{1}{x}.

      2. ddx2x=2xlog⁡(2)\frac{d}{d x} 2^{x} = 2^{x} \log{\left(2 \right)}

      The result is: 2xlog⁡(2)+1x2^{x} \log{\left(2 \right)} + \frac{1}{x}

    The result is: (1−tan⁡(x))(2xlog⁡(2)+1x)−(2x+log⁡(x))(sin⁡2(x)+cos⁡2(x))cos⁡2(x)\left(1 - \tan{\left(x \right)}\right) \left(2^{x} \log{\left(2 \right)} + \frac{1}{x}\right) - \frac{\left(2^{x} + \log{\left(x \right)}\right) \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}}

  2. Now simplify:

    −x(2x+log⁡(x))+(2cos⁡(2x+π4)+1)(2xxlog⁡(2)+1)2xcos⁡2(x)\frac{- x \left(2^{x} + \log{\left(x \right)}\right) + \frac{\left(\sqrt{2} \cos{\left(2 x + \frac{\pi}{4} \right)} + 1\right) \left(2^{x} x \log{\left(2 \right)} + 1\right)}{2}}{x \cos^{2}{\left(x \right)}}

The answer is:

−x(2x+log⁡(x))+(2cos⁡(2x+π4)+1)(2xxlog⁡(2)+1)2xcos⁡2(x)\frac{- x \left(2^{x} + \log{\left(x \right)}\right) + \frac{\left(\sqrt{2} \cos{\left(2 x + \frac{\pi}{4} \right)} + 1\right) \left(2^{x} x \log{\left(2 \right)} + 1\right)}{2}}{x \cos^{2}{\left(x \right)}}

The graph
02468-8-6-4-2-1010-5000050000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
             /1    x       \   /        2   \ /          x\
(1 - tan(x))*|- + 2 *log(2)| + \-1 - tan (x)/*\log(x) + 2 /
             \x            /
(1−tan⁡(x))(2xlog⁡(2)+1x)+(2x+log⁡(x))(−tan⁡2(x)−1)\left(1 - \tan{\left(x \right)}\right) \left(2^{x} \log{\left(2 \right)} + \frac{1}{x}\right) + \left(2^{x} + \log{\left(x \right)}\right) \left(- \tan^{2}{\left(x \right)} - 1\right)
Simplify
The second derivative [src] 
 /              /  1     x    2   \     /       2   \ /1    x       \     /       2   \ / x         \       \
-|(-1 + tan(x))*|- -- + 2 *log (2)| + 2*\1 + tan (x)/*|- + 2 *log(2)| + 2*\1 + tan (x)/*\2  + log(x)/*tan(x)|
 |              |   2             |                   \x            /                                       |
 \              \  x              /                                                                         /
−(2(2x+log⁡(x))(tan⁡2(x)+1)tan⁡(x)+2(2xlog⁡(2)+1x)(tan⁡2(x)+1)+(2xlog⁡(2)2−1x2)(tan⁡(x)−1))- (2 \left(2^{x} + \log{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 2 \left(2^{x} \log{\left(2 \right)} + \frac{1}{x}\right) \left(\tan^{2}{\left(x \right)} + 1\right) + \left(2^{x} \log{\left(2 \right)}^{2} - \frac{1}{x^{2}}\right) \left(\tan{\left(x \right)} - 1\right))
Simplify
3-я производная [src] 
 /              /2     x    3   \     /       2   \ /  1     x    2   \     /       2   \ /         2   \ / x         \     /       2   \ /1    x       \       \
-|(-1 + tan(x))*|-- + 2 *log (2)| + 3*\1 + tan (x)/*|- -- + 2 *log (2)| + 2*\1 + tan (x)/*\1 + 3*tan (x)/*\2  + log(x)/ + 6*\1 + tan (x)/*|- + 2 *log(2)|*tan(x)|
 |              | 3             |                   |   2             |                                                                   \x            /       |
 \              \x              /                   \  x              /                                                                                         /
−(2(2x+log⁡(x))(tan⁡2(x)+1)(3tan⁡2(x)+1)+6(2xlog⁡(2)+1x)(tan⁡2(x)+1)tan⁡(x)+3(2xlog⁡(2)2−1x2)(tan⁡2(x)+1)+(2xlog⁡(2)3+2x3)(tan⁡(x)−1))- (2 \left(2^{x} + \log{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) + 6 \left(2^{x} \log{\left(2 \right)} + \frac{1}{x}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 3 \left(2^{x} \log{\left(2 \right)}^{2} - \frac{1}{x^{2}}\right) \left(\tan^{2}{\left(x \right)} + 1\right) + \left(2^{x} \log{\left(2 \right)}^{3} + \frac{2}{x^{3}}\right) \left(\tan{\left(x \right)} - 1\right))
Simplify
The third derivative [src] 
 /              /2     x    3   \     /       2   \ /  1     x    2   \     /       2   \ /         2   \ / x         \     /       2   \ /1    x       \       \
-|(-1 + tan(x))*|-- + 2 *log (2)| + 3*\1 + tan (x)/*|- -- + 2 *log (2)| + 2*\1 + tan (x)/*\1 + 3*tan (x)/*\2  + log(x)/ + 6*\1 + tan (x)/*|- + 2 *log(2)|*tan(x)|
 |              | 3             |                   |   2             |                                                                   \x            /       |
 \              \x              /                   \  x              /                                                                                         /
−(2(2x+log⁡(x))(tan⁡2(x)+1)(3tan⁡2(x)+1)+6(2xlog⁡(2)+1x)(tan⁡2(x)+1)tan⁡(x)+3(2xlog⁡(2)2−1x2)(tan⁡2(x)+1)+(2xlog⁡(2)3+2x3)(tan⁡(x)−1))- (2 \left(2^{x} + \log{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) + 6 \left(2^{x} \log{\left(2 \right)} + \frac{1}{x}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 3 \left(2^{x} \log{\left(2 \right)}^{2} - \frac{1}{x^{2}}\right) \left(\tan^{2}{\left(x \right)} + 1\right) + \left(2^{x} \log{\left(2 \right)}^{3} + \frac{2}{x^{3}}\right) \left(\tan{\left(x \right)} - 1\right))
Simplify
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