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The derivative of the function/ e^x(sinx-cosx)

Derivative of e^x(sinx-cosx)

The solution

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

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

d / x                  \
--\e *(sin(x) - cos(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 sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    So, the result is: $\sin{\left(x \right)}$
  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  + (sin(x) - cos(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 e^x(sinx-cosx)