**High School Number Sense Lesson 74: Derivative of a Function**

Let's dip a little into Calculus today, shall we? We'll start with one of the two basic calculations in Calculus: the

**derivative**. The process of finding the

**derivative**of a function is called

**differentiation**. This concept appeared

**12 times**last year, with a median placement at

**question # 74**.

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**Definition:**

The derivative of a function is often described as the

**instantaneous rate of change**of the function. According to Wikipedia (the great treasure of crowdsourced knowledge...): "The derivative of a function

**y = f(x)**of a variable

**x**is a measure of the rate at which the value

**y**of the function changes with respect to the change of the variable

**x**." Confused? You may want to spend some time on

**Khan Academy's**lessons on derivatives. In the meantime, let's try to do some differentiation.

**How to Calculate:**

The derivative of a

**variable**is the product of:

- the
*exponent*of the variable, and - the
*variable*taken to the power of (the exponent minus 1).

**constant**is 0.

In other words:

**x**is

^{r}**r(x**.

^{r-1})The derivative of

**11**(or any other constant) is

**0**.

So, if given the function

**f(x) = x**,

^{5}- 2x^{4}+ 3x^{3}- 4x^{2}+ 5x - 61. The derivative of

**x**is

^{5}**5x**.

^{4}2. The derivative of

**-2x**is

^{4}**-8x**.

^{3}3. The derivative of

**3x**is

^{3}**9x**.

^{2}4. The derivative of

**-4x**is

^{2}**-8x**.

5. The derivative of

**5x**is

**5**.

6. The derivative of

**-6**is

**0**.

So the derivative of the function is

**5x**.

^{4}- 8x^{3}+ 9x^{2}- 8x + 5**f'(x)**, where

**f'(x)**is the derivative of

**f(x)**. Also,

**f''(x)**is the derivative of

**f'(x)**, and so on. Let's look at some examples:

**Example 1:**

**If f(x) = 3x**

^{2}- 4x + 2, then f'(5) = ___1. Find the derivative of the function: f'(x) = 2(3x

^{2-1}) - 4x

^{1-0}= 6x - 4x

^{0}=

**6x - 4**.

2. Substitute

**5**for x: f'(5) = 6(5) - 4 = 30 - 4 =

**26**.

**Example 2:**

**If f(x) = 2x**

^{3}- 6, then f'(-1) = ___1. Find the derivative of the function:

**f'(x) = 6x**.

^{2}2. Substitute

**-1**for x: f'(-1) = 6(-1)

^{2}= 6(1) =

**6**.

**Example 3:**

**If f(x) = x**

^{3}+ 2x^{2}- x - 2, find f'(2). ___1. Find the derivative of the function:

**f'(x) = 3x**.

^{2}+ 4x - 12. Substitute

**2**for x: f'(2) = 3(2

^{2}) + 4(2) + 1 = 3(4) + 8 + 1 = 12 + 9 =

**21**.

**Example 4:**

**If f(x) = x**

^{3}+ 2x^{2}+ 3x + 4, find f''(5). ___1. Find the derivative of the function:

**f'(x) = 3x**.

^{2}+ 4x + 32. Find the second derivative (the derivative of that derivative):

**f''(x) = 6x + 4**.

3. Substitute

**5**for x: f''(5) = 6(5) + 4 = 30 + 4 =

**34**.

**Example 5:**

**f(x) = x**

^{3}- 3x^{2}- 5x + 7. Find f''(-1). ___1. Find the derivative of the function:

**f'(x) = 3x**.

^{2}- 6x - 52. Find the 2nd derivative:

**f''(x) = 6x - 6**.

3. Substitute

**-1**for x: f''(-1) = 6(-1) - 6 = -6 - 6 =

**-12**.

**Check back soon for a free worksheet to help you practice FunctionDeriv.**

**Up Next for High School: SlopeLineTang**