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Derivative of:
Derivative of x*e
Derivative of sqrt(x^3)
Derivative of e^(x*(-3))
Derivative of 7
Sum of series:
f(x)
Integral of d{x}:
f(x)
f(x)
Identical expressions
f(x)=x^3cos(x)-5x^ two - one
f(x) equally x cubed co sinus of e of (x) minus 5x squared minus 1
f(x) equally x cubed co sinus of e of (x) minus 5x to the power of two minus one
f(x)=x3cos(x)-5x2-1
fx=x3cosx-5x2-1
f(x)=x³cos(x)-5x²-1
f(x)=x to the power of 3cos(x)-5x to the power of 2-1
fx=x^3cosx-5x^2-1
Similar expressions
f(x)=x^3cos(x)+5x^2-1
f(x)=x^3cos(x)-5x^2+1
f(x)=x^3cosx-5x^2-1

Derivative of f(x)=x^3cos(x)-5x^2-1

You have entered [src]

 3             2    
x *cos(x) - 5*x  - 1

$$\left(x^{3} \cos{\left(x \right)} - 5 x^{2}\right) - 1$$

Transform

Detail solution

Differentiate $\left(x^{3} \cos{\left(x \right)} - 5 x^{2}\right) - 1$ term by term:
1. Differentiate $x^{3} \cos{\left(x \right)} - 5 x^{2}$ term by term:
  1. Apply the product rule:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = x^{3}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
    $g{\left(x \right)} = \cos{\left(x \right)}$ ; 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: $- x^{3} \sin{\left(x \right)} + 3 x^{2} \cos{\left(x \right)}$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    So, the result is: $- 10 x$
  The result is: $- x^{3} \sin{\left(x \right)} + 3 x^{2} \cos{\left(x \right)} - 10 x$
2. The derivative of the constant $-1$ is zero.
The result is: $- x^{3} \sin{\left(x \right)} + 3 x^{2} \cos{\left(x \right)} - 10 x$
Now simplify:

$x \left(- x^{2} \sin{\left(x \right)} + 3 x \cos{\left(x \right)} - 10\right)$

The answer is:

$x \left(- x^{2} \sin{\left(x \right)} + 3 x \cos{\left(x \right)} - 10\right)$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

         3             2       
-10*x - x *sin(x) + 3*x *cos(x)

$$- x^{3} \sin{\left(x \right)} + 3 x^{2} \cos{\left(x \right)} - 10 x$$

Simplify

The second derivative [src]

       3             2                    
-10 - x *cos(x) - 6*x *sin(x) + 6*x*cos(x)

$$- x^{3} \cos{\left(x \right)} - 6 x^{2} \sin{\left(x \right)} + 6 x \cos{\left(x \right)} - 10$$

Simplify

The third derivative [src]

            3                           2       
6*cos(x) + x *sin(x) - 18*x*sin(x) - 9*x *cos(x)

$$x^{3} \sin{\left(x \right)} - 9 x^{2} \cos{\left(x \right)} - 18 x \sin{\left(x \right)} + 6 \cos{\left(x \right)}$$

Simplify

The graph