---
title: "3.6.2 Derivatives"
layout: ../../../../layouts/LessonLayout.astro
---

import { Question } from "../../../../components/Question";

# 3.6.2 Derivatives

Usually on the number sense test, there is guaranteed to be a derivative (or double derivative) of a polynomial. Almost every single time, the use of the power rule is all that is required, so let's see how we can take the derivative of a polynomial:

Define: $f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x^1 + a_0 x^0$ then

$$
f'(x) = a_n(n)x^{n-1} + a_{n-1}(n - 1)x^{n-2} + \dots + a_1(1)x^0
$$

So the procedure is you multiply the coefficient by the power and then lower the power (notice that a constant after differentiating disappears). Let's look at an example:

**Problem:** Let $f(x) = x^3 - 3x^2 + x - 3$, solve for $f'(2)$.
**Solution:** $f'(x) = 1 \cdot 3x^2 - 3 \cdot 2x + 1 \Rightarrow f'(2) = 1 \cdot 3 \cdot 2^2 - 3 \cdot 2 \cdot 2 + 1 = 1$

When approached with taking double derivatives ($f''(x)$), then just follow the procedure twice:

**Problem:** Let $f(x) = 5x^3 + 3x^2 - 7$, solve for $f''(1)$.
**Solution:** $f'(x) = 5 \cdot 3x^2 + 3 \cdot 2x = 15x^2 + 6x \Rightarrow f''(x) = 15 \cdot 2x + 6 \Rightarrow f''(1) = 30 \cdot 1 + 6 = 36$

In the off chance that the derivative of sine/cosine or the $e^x/\ln x$ is needed (like for using l'hˆopitals rule), here is a chart showing these functions and their derivatives:

| Function | Derivative |
| :------- | :--------- |
| $\sin x$ | $\cos x$   |
| $\cos x$ | $-\sin x$  |
| $e^x$    | $e^x$      |
| $\ln x$  | $1/x$      |

For more derivative rules, consult a calculus textbook (it would be good to be familiar with more derivative rules for the math test, but unlikely those rules will be applied to the number sense test).

Here are some problems to practice taking derivatives:

### Problem Set 3.6.2

<div className="not-prose grid grid-cols-2 md:grid-cols-3 gap-2 my-4">
  <Question type="math" id="derivatives-1" question="$f(x) = 3x^2 + x - 5, f'(-2) =$" answer="-11" client:load/>
  <Question type="math" id="derivatives-2" question="$f(x) = x^2 - 2x + 22, f'(2) =$" answer="2" client:load/>
  <Question type="math" id="derivatives-3" question="$g(x) = 2x^2 - 3x + 1, g'(2) =$" answer="5" client:load/>
  <Question type="math" id="derivatives-4" question="$f(x) = 3x^3 - 3x + 3, f'(-3) =$" answer="78" client:load/>
  <Question type="math" id="derivatives-5" question="$f(x) = 4x^3 + 2x^2, f''(-.5) =$" answer="-22" client:load/>
  <Question type="math" id="derivatives-6" question="$f(x) = x^3 - 3x + 3, f'(3) =$" answer="24" client:load/>
  <Question type="math" id="derivatives-7" question="$f(x) = x^4 - 4x + 4, f'(4) =$" answer="252" client:load/>
  <Question type="math" id="derivatives-8" question="$f(x) = 3x^2 + 4x - 5, f'(-6) =$" answer="-32" client:load/>
  <Question type="math" id="derivatives-9" question="$f(x) = 2x^3 - 3x^4, f''(-1) =$" answer="48" client:load/>
  <Question type="math" id="derivatives-10" question="$f(x) = 4x^3 - 3x^2 + 1, f'(-1) =$" answer="-6" client:load/>
  <Question type="math" id="derivatives-11" question="$f(x) = x^2 - 3x + 4, f''(-1) =$" answer="2" client:load/>
  <Question type="math" id="derivatives-12" question="$f(x) = 3x + 5x^2 - 7x^4, f'(1) =$" answer="-15" client:load/>
  <Question type="math" id="derivatives-13" question="$f(x) = 3x^3 - 2x^2 + x, f''(1) =$" answer="14" client:load/>
  <Question type="math" id="derivatives-14" question="$f(x) = 2x^3 - 4x^2 + 6x, f'(1) =$" answer="2" client:load/>
  <Question type="math" id="derivatives-15" question="$f(x) = x^5 + x^3 - x, f''(2) =$" answer="64" client:load/>
  <Question type="math" id="derivatives-16" question="$f(x) = 4x^3 - 3x^2 + x, f'(-1) =$" answer="-15" client:load/>
  <Question type="math" id="derivatives-17" question="$f(x) = x^3 - 3x^2 + 5x, f''(2) =$" answer="6" client:load/>
  <Question type="math" id="derivatives-18" question="$f(x) = 4x^3 - 3x^2 + 2x, f''(1) =$" answer="18" client:load/>
  <Question type="math" id="derivatives-19" question="$f(x) = 2x^2 - 3x + 4, f'(-1) =$" answer="-7" client:load/>
  <Question type="math" id="derivatives-20" question="$f(x) = 4 - 3x - 2x^2, f'(-1) =$" answer="1" client:load/>
  <Question type="math" id="derivatives-21" question="$g(x) = x^3 - 3x - 3, g'(-3) =$" answer="24" client:load/>
  <Question type="math" id="derivatives-22" question="$g(x) = 2x^3 + 3x^2 + 5, g''(4) =$" answer="60" client:load/>
  <Question type="math" id="derivatives-23" question="$h(x) = 1 + 2x^2 - 3x^3, h''(4) =$" answer="-68" client:load/>
  <Question type="math" id="derivatives-24" question="$f(x) = 4 - 3x^2 + 2x^3, f''(5) =$" answer="54" client:load/>
  <Question type="math" id="derivatives-25" question="$f(x) = x^3 - 3x + 3, f'(-3) =$" answer="24" client:load/>
  <Question type="math" id="derivatives-26" question="$f(x) = x^4 - 4x^2 + 4, f'(-4) =$" answer="-288" client:load/>
  <Question type="math" id="derivatives-27" question="$f(x) = 3x^3 + 3x - 3, f'(-3) =$" answer="84" client:load/>
  <Question type="math" id="derivatives-28" question="$f(x) = 3x^2 - 4x + 2, f'(1/3) =$" answer="-2" client:load/>
</div>