Mister Exam

Lang:

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

Derivative of e^(x)(sinx+cosx)

The solution

You have entered [src]

 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 cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  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:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The graph

Derivative of e^(x)(sinx+cosx)