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The derivative of the function/ y=x^5(x+1)

Derivative of y=x^5(x+1)

The solution

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 5        
x *(x + 1)

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

d / 5        \
--\x *(x + 1)/
dx

$$\frac{d}{d x} x^{5} \left(x + 1\right)$$

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x^{5}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{5}$ goes to $5 x^{4}$
$g{\left(x \right)} = x + 1$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x + 1$ term by term:
  1. Apply the power rule: $x$ goes to $1$
  2. The derivative of the constant $1$ is zero.
  The result is: $1$
The result is: $x^{5} + 5 x^{4} \left(x + 1\right)$
Now simplify:

$x^{4} \cdot \left(6 x + 5\right)$

The answer is:

$x^{4} \cdot \left(6 x + 5\right)$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 5      4        
x  + 5*x *(x + 1)

$$x^{5} + 5 x^{4} \left(x + 1\right)$$

Simplify

The second derivative [src]

    3          
10*x *(2 + 3*x)

$$10 x^{3} \cdot \left(3 x + 2\right)$$

Simplify

The third derivative [src]

    2          
60*x *(1 + 2*x)

$$60 x^{2} \cdot \left(2 x + 1\right)$$

Simplify

The graph

Derivative of y=x^5(x+1)