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е^x*(cos(x)-sin(x))

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Identical expressions
е^x*(cos(x)-sin(x))
е to the power of x multiply by ( co sinus of e of (x) minus sinus of (x))
еx*(cos(x)-sin(x))
еx*cosx-sinx
е^x(cos(x)-sin(x))
еx(cos(x)-sin(x))
еxcosx-sinx
е^xcosx-sinx
Similar expressions
е^x*(cos(x)+sin(x))
е^x*(cosx-sinx)

The derivative of the function/ е^x*(cos(x)-sin(x))

Derivative of е^x*(cos(x)-sin(x))

The solution

You have entered [src]

 x                  
e *(cos(x) - sin(x))

$$\left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x}$$

d / x                  \
--\e *(cos(x) - sin(x))/
dx

$$\frac{d}{d x} \left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x}$$

Transform

Detail solution

Apply the product rule:

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

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

$- 2 e^{x} \sin{\left(x \right)}$

The answer is:

$- 2 e^{x} \sin{\left(x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                    x                      x
(-cos(x) - sin(x))*e  + (cos(x) - sin(x))*e

$$\left(- \sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{x} + \left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x}$$

Simplify

The second derivative [src]

                      x
-2*(cos(x) + sin(x))*e

$$- 2 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x}$$

Simplify

The third derivative [src]

           x
-4*cos(x)*e

$$- 4 e^{x} \cos{\left(x \right)}$$

Simplify

The graph

Derivative of е^x*(cos(x)-sin(x))