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Identical expressions
y=(one -tgx)*(lnx+ two ^x)
y equally (1 minus tgx) multiply by (lnx plus 2 to the power of x)
y equally (one minus tgx) multiply by (lnx plus two to the power of x)
y=(1-tgx)*(lnx+2x)
y=1-tgx*lnx+2x
y=(1-tgx)(lnx+2^x)
y=(1-tgx)(lnx+2x)
y=1-tgxlnx+2x
y=1-tgxlnx+2^x
Similar expressions
y=(1+tgx)*(lnx+2^x)
y=(1-tgx)*(lnx-2^x)

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

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

The solution

You have entered [src]

             /          x\
(1 - tan(x))*\log(x) + 2 /

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

(1 - tan(x))*(log(x) + 2^x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = 1 - \tan{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $1 - \tan{\left(x \right)}$ term by term:
  1. The derivative of the constant $1$ 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:
      
      $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
    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: $- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  The result is: $- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
$g{\left(x \right)} = 2^{x} + \log{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $2^{x} + \log{\left(x \right)}$ term by term:
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  2. $\frac{d}{d x} 2^{x} = 2^{x} \log{\left(2 \right)}$
  The result is: $2^{x} \log{\left(2 \right)} + \frac{1}{x}$
The result is: $\left(1 - \tan{\left(x \right)}\right) \left(2^{x} \log{\left(2 \right)} + \frac{1}{x}\right) - \frac{\left(2^{x} + \log{\left(x \right)}\right) \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}}$
Now simplify:

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

The answer is:

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

The graph

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            /

$$\left(1 - \tan{\left(x \right)}\right) \left(2^{x} \log{\left(2 \right)} + \frac{1}{x}\right) + \left(2^{x} + \log{\left(x \right)}\right) \left(- \tan^{2}{\left(x \right)} - 1\right)$$

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              /                                                                         /

$$- (2 \left(2^{x} + \log{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 2 \left(2^{x} \log{\left(2 \right)} + \frac{1}{x}\right) \left(\tan^{2}{\left(x \right)} + 1\right) + \left(2^{x} \log{\left(2 \right)}^{2} - \frac{1}{x^{2}}\right) \left(\tan{\left(x \right)} - 1\right))$$

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              /                                                                                         /

$$- (2 \left(2^{x} + \log{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) + 6 \left(2^{x} \log{\left(2 \right)} + \frac{1}{x}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 3 \left(2^{x} \log{\left(2 \right)}^{2} - \frac{1}{x^{2}}\right) \left(\tan^{2}{\left(x \right)} + 1\right) + \left(2^{x} \log{\left(2 \right)}^{3} + \frac{2}{x^{3}}\right) \left(\tan{\left(x \right)} - 1\right))$$

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              /                                                                                         /

Simplify

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Identical expressions

Similar expressions

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

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The solution