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Identical expressions
(-secx+ one)exp(x)cosx
( minus secx plus 1) exponent of (x) co sinus of e of x
( minus secx plus one) exponent of (x) co sinus of e of x
-secx+1expxcosx
Similar expressions
(-secx-1)exp(x)cosx
(secx+1)exp(x)cosx

The derivative of the function/ (-secx+1)exp(x)cosx

Derivative of (-secx+1)exp(x)cosx

The solution

You have entered [src]

               x       
(-sec(x) + 1)*e *cos(x)

$$\left(1 - \sec{\left(x \right)}\right) e^{x} \cos{\left(x \right)}$$

((-sec(x) + 1)*exp(x))*cos(x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \left(1 - \sec{\left(x \right)}\right) e^{x}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = 1 - \sec{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Differentiate $1 - \sec{\left(x \right)}$ term by term:
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Rewrite the function to be differentiated:
        
        $\sec{\left(x \right)} = \frac{1}{\cos{\left(x \right)}}$
      2. Let $u = \cos{\left(x \right)}$ .
      3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
      4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
        
        The derivative of cosine is negative sine:
        
        $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
        
        The result of the chain rule is:
        
        $\frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
      So, the result is: $- \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
    2. The derivative of the constant $1$ is zero.
    The result is: $- \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  $g{\left(x \right)} = e^{x}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of $e^{x}$ is itself.
  The result is: $\left(1 - \sec{\left(x \right)}\right) e^{x} - \frac{e^{x} \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
$g{\left(x \right)} = \cos{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of cosine is negative sine:
  
  $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
The result is: $- \left(1 - \sec{\left(x \right)}\right) e^{x} \sin{\left(x \right)} + \left(\left(1 - \sec{\left(x \right)}\right) e^{x} - \frac{e^{x} \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right) \cos{\left(x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

/               x    x              \                         x       
\(-sec(x) + 1)*e  - e *sec(x)*tan(x)/*cos(x) - (-sec(x) + 1)*e *sin(x)

$$- \left(1 - \sec{\left(x \right)}\right) e^{x} \sin{\left(x \right)} + \left(\left(1 - \sec{\left(x \right)}\right) e^{x} - e^{x} \tan{\left(x \right)} \sec{\left(x \right)}\right) \cos{\left(x \right)}$$

Simplify

The second derivative [src]

/                       /     /         2   \                                  \                                                \  x
\(-1 + sec(x))*cos(x) - \-1 + \1 + 2*tan (x)/*sec(x) + 2*sec(x)*tan(x) + sec(x)/*cos(x) + 2*(-1 + sec(x)*tan(x) + sec(x))*sin(x)/*e

$$\left(\left(\sec{\left(x \right)} - 1\right) \cos{\left(x \right)} + 2 \left(\tan{\left(x \right)} \sec{\left(x \right)} + \sec{\left(x \right)} - 1\right) \sin{\left(x \right)} - \left(\left(2 \tan^{2}{\left(x \right)} + 1\right) \sec{\left(x \right)} + 2 \tan{\left(x \right)} \sec{\left(x \right)} + \sec{\left(x \right)} - 1\right) \cos{\left(x \right)}\right) e^{x}$$

Simplify

The third derivative [src]

/                        /       /         2   \                            /         2   \                       \                                                     /     /         2   \                                  \       \  x
\-(-1 + sec(x))*sin(x) - \-1 + 3*\1 + 2*tan (x)/*sec(x) + 3*sec(x)*tan(x) + \5 + 6*tan (x)/*sec(x)*tan(x) + sec(x)/*cos(x) + 3*(-1 + sec(x)*tan(x) + sec(x))*cos(x) + 3*\-1 + \1 + 2*tan (x)/*sec(x) + 2*sec(x)*tan(x) + sec(x)/*sin(x)/*e

$$\left(- \left(\sec{\left(x \right)} - 1\right) \sin{\left(x \right)} + 3 \left(\tan{\left(x \right)} \sec{\left(x \right)} + \sec{\left(x \right)} - 1\right) \cos{\left(x \right)} + 3 \left(\left(2 \tan^{2}{\left(x \right)} + 1\right) \sec{\left(x \right)} + 2 \tan{\left(x \right)} \sec{\left(x \right)} + \sec{\left(x \right)} - 1\right) \sin{\left(x \right)} - \left(3 \left(2 \tan^{2}{\left(x \right)} + 1\right) \sec{\left(x \right)} + \left(6 \tan^{2}{\left(x \right)} + 5\right) \tan{\left(x \right)} \sec{\left(x \right)} + 3 \tan{\left(x \right)} \sec{\left(x \right)} + \sec{\left(x \right)} - 1\right) \cos{\left(x \right)}\right) e^{x}$$

Simplify

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Similar expressions

Derivative of (-secx+1)exp(x)cosx

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Piecewise:

The solution