Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of (-2)/x^3
Derivative of e^(3/x)
Derivative of (2*x+1)*(x^2+3*x-1)
Derivative of (x*3)^(4-2*x)
Identical expressions
exp(cos(t)- two *sin(t)+ two)
exponent of ( co sinus of e of (t) minus 2 multiply by sinus of (t) plus 2)
exponent of ( co sinus of e of (t) minus two multiply by sinus of (t) plus two)
exp(cos(t)-2sin(t)+2)
expcost-2sint+2
Similar expressions
exp(cos(t)-2*sin(t)-2)
exp(cos(t)+2*sin(t)+2)

The derivative of the function/ exp(cos(t)-2*sin(t)+2)

Derivative of exp(cos(t)-2*sin(t)+2)

The solution

You have entered [src]

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

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

exp(cos(t) - 2*sin(t) + 2)

Detail solution

Let $u = \left(- 2 \sin{\left(t \right)} + \cos{\left(t \right)}\right) + 2$ .
The derivative of $e^{u}$ is itself.
Then, apply the chain rule. Multiply by $\frac{d}{d t} \left(\left(- 2 \sin{\left(t \right)} + \cos{\left(t \right)}\right) + 2\right)$ :
1. Differentiate $\left(- 2 \sin{\left(t \right)} + \cos{\left(t \right)}\right) + 2$ term by term:
  1. Differentiate $- 2 \sin{\left(t \right)} + \cos{\left(t \right)}$ term by term:
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d t} \cos{\left(t \right)} = - \sin{\left(t \right)}$
    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:
        
        $\frac{d}{d t} \sin{\left(t \right)} = \cos{\left(t \right)}$
      So, the result is: $- 2 \cos{\left(t \right)}$
    The result is: $- \sin{\left(t \right)} - 2 \cos{\left(t \right)}$
  2. The derivative of the constant $2$ is zero.
  The result is: $- \sin{\left(t \right)} - 2 \cos{\left(t \right)}$
The result of the chain rule is:

$\left(- \sin{\left(t \right)} - 2 \cos{\left(t \right)}\right) e^{\left(- 2 \sin{\left(t \right)} + \cos{\left(t \right)}\right) + 2}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

/                   2                    \  2 - 2*sin(t) + cos(t)
\(2*cos(t) + sin(t))  - cos(t) + 2*sin(t)/*e

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

Simplify

The third derivative [src]

                    /                       2                      \  2 - 2*sin(t) + cos(t)
(2*cos(t) + sin(t))*\1 - (2*cos(t) + sin(t))  - 6*sin(t) + 3*cos(t)/*e

$$\left(\sin{\left(t \right)} + 2 \cos{\left(t \right)}\right) \left(- \left(\sin{\left(t \right)} + 2 \cos{\left(t \right)}\right)^{2} - 6 \sin{\left(t \right)} + 3 \cos{\left(t \right)} + 1\right) e^{- 2 \sin{\left(t \right)} + \cos{\left(t \right)} + 2}$$

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of exp(cos(t)-2*sin(t)+2)

The graph:

Piecewise:

The solution