Derivative of y=x⁸+7x⁷-6x⁶+5x⁵-4x⁴-x³-x²+5

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 8      7      6      5      4    3    2    
x  + 7*x  - 6*x  + 5*x  - 4*x  - x  - x  + 5

$$x^{8} + 7 x^{7} - 6 x^{6} + 5 x^{5} - 4 x^{4} - x^{3} - x^{2} + 5$$

d / 8      7      6      5      4    3    2    \
--\x  + 7*x  - 6*x  + 5*x  - 4*x  - x  - x  + 5/
dx

$$\frac{d}{d x} \left(x^{8} + 7 x^{7} - 6 x^{6} + 5 x^{5} - 4 x^{4} - x^{3} - x^{2} + 5\right)$$

Transform

Detail solution

Differentiate $x^{8} + 7 x^{7} - 6 x^{6} + 5 x^{5} - 4 x^{4} - x^{3} - x^{2} + 5$ term by term:
1. Apply the power rule: $x^{8}$ goes to $8 x^{7}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $x^{7}$ goes to $7 x^{6}$
  So, the result is: $49 x^{6}$
3. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{6}$ goes to $6 x^{5}$
    So, the result is: $36 x^{5}$
  So, the result is: $- 36 x^{5}$
4. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $x^{5}$ goes to $5 x^{4}$
  So, the result is: $25 x^{4}$
5. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{4}$ goes to $4 x^{3}$
    So, the result is: $16 x^{3}$
  So, the result is: $- 16 x^{3}$
6. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
  So, the result is: $- 3 x^{2}$
7. 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: $- 2 x$
8. The derivative of the constant $5$ is zero.
The result is: $8 x^{7} + 49 x^{6} - 36 x^{5} + 25 x^{4} - 16 x^{3} - 3 x^{2} - 2 x$
Now simplify:

$x \left(8 x^{6} + 49 x^{5} - 36 x^{4} + 25 x^{3} - 16 x^{2} - 3 x - 2\right)$

The answer is:

$x \left(8 x^{6} + 49 x^{5} - 36 x^{4} + 25 x^{3} - 16 x^{2} - 3 x - 2\right)$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      5       3      2            7       4       6
- 36*x  - 16*x  - 3*x  - 2*x + 8*x  + 25*x  + 49*x

$$8 x^{7} + 49 x^{6} - 36 x^{5} + 25 x^{4} - 16 x^{3} - 3 x^{2} - 2 x$$

Simplify

The second derivative [src]

  /         4       2             6       3        5\
2*\-1 - 90*x  - 24*x  - 3*x + 28*x  + 50*x  + 147*x /

$$2 \cdot \left(28 x^{6} + 147 x^{5} - 90 x^{4} + 50 x^{3} - 24 x^{2} - 3 x - 1\right)$$

Simplify

The third derivative [src]

  /          3              2       5        4\
6*\-1 - 120*x  - 16*x + 50*x  + 56*x  + 245*x /

$$6 \cdot \left(56 x^{5} + 245 x^{4} - 120 x^{3} + 50 x^{2} - 16 x - 1\right)$$

Simplify

The graph

Derivative of y=x⁸+7x⁷-6x⁶+5x⁵-4x⁴-x³-x²+5

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Identical expressions

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Derivative of y=x⁸+7x⁷-6x⁶+5x⁵-4x⁴-x³-x²+5

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The solution