1. The basics Consider the function $$ f(x)=1+x^{2}+x^{3}+x^{4}+x^{5}+x^{6} $$ a. What is its rate of change when $$ x $$ is 0 ? b. What is its rate of change when $$ x $$ is 1 ? c. What is its second derivative? d. What is its fifth derivative?

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**Step 1. Write down the function**

We have
$$
f(x) = 1 + x^2 + x^3 + x^4 + x^5 + x^6.
$$

**Step 2. Compute the first derivative $$ f'(x) $$**

Differentiate each term:
- Derivative of $$1$$ is $$0$$.
- Derivative of $$x^2$$ is $$2x$$.
- Derivative of $$x^3$$ is $$3x^2$$.
- Derivative of $$x^4$$ is $$4x^3$$.
- Derivative of $$x^5$$ is $$5x^4$$.
- Derivative of $$x^6$$ is $$6x^5$$.

Thus,
$$
f'(x) = 2x + 3x^2 + 4x^3 + 5x^4 + 6x^5.
$$

**(a) Rate of change when $$ x = 0 $$**

Evaluate $$ f'(x) $$ at $$ x = 0 $$:
$$
f'(0) = 2(0) + 3(0)^2 + 4(0)^3 + 5(0)^4 + 6(0)^5 = 0.
$$

**(b) Rate of change when $$ x = 1 $$**

Evaluate $$ f'(x) $$ at $$ x = 1 $$:
$$
f'(1) = 2(1) + 3(1)^2 + 4(1)^3 + 5(1)^4 + 6(1)^5 = 2 + 3 + 4 + 5 + 6 = 20.
$$

**Step 3. Compute the second derivative $$ f''(x) $$**

Differentiate $$ f'(x) = 2x + 3x^2 + 4x^3 + 5x^4 + 6x^5 $$:
- Derivative of $$2x$$ is $$2$$.
- Derivative of $$3x^2$$ is $$6x$$.
- Derivative of $$4x^3$$ is $$12x^2$$.
- Derivative of $$5x^4$$ is $$20x^3$$.
- Derivative of $$6x^5$$ is $$30x^4$$.

Thus,
$$
f''(x) = 2 + 6x + 12x^2 + 20x^3 + 30x^4.
$$

**(c) The second derivative is**
$$
f''(x) = 2 + 6x + 12x^2 + 20x^3 + 30x^4.
$$

**Step 4. Compute the fifth derivative $$ f^{(5)}(x) $$**

We calculate the derivatives term by term from the original $$ f(x) $$:

- For $$1$$: All derivatives are $$0$$ (for orders $$ \ge 1 $$).

- For $$x^2$$:
  - $$ \frac{d}{dx} x^2 = 2x $$
  - $$ \frac{d^2}{dx^2} x^2 = 2 $$
  - Higher derivatives become $$0$$.

- For $$x^3$$:
  - $$ \frac{d}{dx} x^3 = 3x^2 $$
  - $$ \frac{d^2}{dx^2} x^3 = 6x $$
  - $$ \frac{d^3}{dx^3} x^3 = 6 $$
  - $$ \frac{d^4}{dx^4} x^3 = 0 $$
  - Subsequent derivatives are $$0$$.

- For $$x^4$$:
  - $$ \frac{d}{dx} x^4 = 4x^3 $$
  - $$ \frac{d^2}{dx^2} x^4 = 12x^2 $$
  - $$ \frac{d^3}{dx^3} x^4 = 24x $$
  - $$ \frac{d^4}{dx^4} x^4 = 24 $$
  - $$ \frac{d^5}{dx^5} x^4 = 0 $$.

- For $$x^5$$:
  - $$ \frac{d}{dx} x^5 = 5x^4 $$
  - $$ \frac{d^2}{dx^2} x^5 = 20x^3 $$
  - $$ \frac{d^3}{dx^3} x^5 = 60x^2 $$
  - $$ \frac{d^4}{dx^4} x^5 = 120x $$
  - $$ \frac{d^5}{dx^5} x^5 = 120 $$.

- For $$x^6$$:
  - $$ \frac{d}{dx} x^6 = 6x^5 $$
  - $$ \frac{d^2}{dx^2} x^6 = 30x^4 $$
  - $$ \frac{d^3}{dx^3} x^6 = 120x^3 $$
  - $$ \frac{d^4}{dx^4} x^6 = 360x^2 $$
  - $$ \frac{d^5}{dx^5} x^6 = 720x $$
  - $$ \frac{d^6}{dx^6} x^6 = 720 $$.

Now, summing the contributions only from terms that survive to the fifth derivative:

- From $$x^2$$ and $$x^3$$ and $$x^4$$: Their fifth derivatives are $$0$$.
- From $$x^5$$: The fifth derivative is $$120$$.
- From $$x^6$$: The fifth derivative is $$720x$$.

Thus,
$$
f^{(5)}(x) = 120 + 720x.
$$

**(d) The fifth derivative is**
$$
f^{(5)}(x) = 120 + 720x.
$$
**Simplified Answers:** **(a) Rate of change when $$ x = 0 $$:** 0 **(b) Rate of change when $$ x = 1 $$:** 20 **(c) Second derivative:** $$ f''(x) = 2 + 6x + 12x^2 + 20x^3 + 30x^4 $$ **(d) Fifth derivative:** $$ f^{(5)}(x) = 120 + 720x $$

1. The basics Consider the function \( f(x)=1+x^{2}+x^{3}+x^{4}+x^{5}+x^{6} \) a. What is its rate of change when \( x \) is 0 ? b. What is its rate of change when \( x \) is 1 ? c. What is its second derivative? d. What is its fifth derivative?

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