Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of x^2+cos(x)
Derivative of 5^-x
Derivative of x*sin(3*x)
Derivative of cos(3)^(2)*x-sin(3)^(2)*x
Identical expressions
exp(sinx)×x^(five)+lg(5x+ one)
exponent of ( sinus of x)×x to the power of (5) plus lg(5x plus 1)
exponent of ( sinus of x)×x to the power of (five) plus lg(5x plus one)
exp(sinx)×x(5)+lg(5x+1)
expsinx×x5+lg5x+1
expsinx×x^5+lg5x+1
Similar expressions
exp(sinx)×x^(5)+lg(5x-1)
exp(sinx)×x^(5)-lg(5x+1)

The derivative of the function/ exp(sinx)×x^(5)+lg(5x+1)

Derivative of exp(sinx)×x^(5)+lg(5x+1)

The solution

You have entered [src]

 sin(x)  5               
e      *x  + log(5*x + 1)

$$x^{5} e^{\sin{\left(x \right)}} + \log{\left(5 x + 1 \right)}$$

exp(sin(x))*x^5 + log(5*x + 1)

Detail solution

Differentiate $x^{5} e^{\sin{\left(x \right)}} + \log{\left(5 x + 1 \right)}$ term by term:
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = e^{\sin{\left(x \right)}}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = \sin{\left(x \right)}$ .
  2. The derivative of $e^{u}$ is itself.
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
    1. The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    The result of the chain rule is:
    
    $e^{\sin{\left(x \right)}} \cos{\left(x \right)}$
  $g{\left(x \right)} = x^{5}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Apply the power rule: $x^{5}$ goes to $5 x^{4}$
  The result is: $x^{5} e^{\sin{\left(x \right)}} \cos{\left(x \right)} + 5 x^{4} e^{\sin{\left(x \right)}}$
2. Let $u = 5 x + 1$ .
3. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(5 x + 1\right)$ :
  1. Differentiate $5 x + 1$ term by term:
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $5$
    2. The derivative of the constant $1$ is zero.
    The result is: $5$
  The result of the chain rule is:
  
  $\frac{5}{5 x + 1}$
The result is: $x^{5} e^{\sin{\left(x \right)}} \cos{\left(x \right)} + 5 x^{4} e^{\sin{\left(x \right)}} + \frac{5}{5 x + 1}$
Now simplify:

$\frac{x^{4} \left(5 x + 1\right) \left(x \cos{\left(x \right)} + 5\right) e^{\sin{\left(x \right)}} + 5}{5 x + 1}$

The answer is:

$\frac{x^{4} \left(5 x + 1\right) \left(x \cos{\left(x \right)} + 5\right) e^{\sin{\left(x \right)}} + 5}{5 x + 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   5         4  sin(x)    5         sin(x)
------- + 5*x *e       + x *cos(x)*e      
5*x + 1

$$x^{5} e^{\sin{\left(x \right)}} \cos{\left(x \right)} + 5 x^{4} e^{\sin{\left(x \right)}} + \frac{5}{5 x + 1}$$

Simplify

The second derivative [src]

      25           3  sin(x)    5    2     sin(x)    5  sin(x)              4         sin(x)
- ---------- + 20*x *e       + x *cos (x)*e       - x *e      *sin(x) + 10*x *cos(x)*e      
           2                                                                                
  (1 + 5*x)

$$- x^{5} e^{\sin{\left(x \right)}} \sin{\left(x \right)} + x^{5} e^{\sin{\left(x \right)}} \cos^{2}{\left(x \right)} + 10 x^{4} e^{\sin{\left(x \right)}} \cos{\left(x \right)} + 20 x^{3} e^{\sin{\left(x \right)}} - \frac{25}{\left(5 x + 1\right)^{2}}$$

Simplify

The third derivative [src]

   250           2  sin(x)    5    3     sin(x)    5         sin(x)       4  sin(x)              4    2     sin(x)       3         sin(x)      5         sin(x)       
---------- + 60*x *e       + x *cos (x)*e       - x *cos(x)*e       - 15*x *e      *sin(x) + 15*x *cos (x)*e       + 60*x *cos(x)*e       - 3*x *cos(x)*e      *sin(x)
         3                                                                                                                                                            
(1 + 5*x)

$$- 3 x^{5} e^{\sin{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)} + x^{5} e^{\sin{\left(x \right)}} \cos^{3}{\left(x \right)} - x^{5} e^{\sin{\left(x \right)}} \cos{\left(x \right)} - 15 x^{4} e^{\sin{\left(x \right)}} \sin{\left(x \right)} + 15 x^{4} e^{\sin{\left(x \right)}} \cos^{2}{\left(x \right)} + 60 x^{3} e^{\sin{\left(x \right)}} \cos{\left(x \right)} + 60 x^{2} e^{\sin{\left(x \right)}} + \frac{250}{\left(5 x + 1\right)^{3}}$$

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of exp(sinx)×x^(5)+lg(5x+1)

The graph:

Piecewise:

The solution