Derivatives

Definition - Derivative

The derivative of a function is the equation that gives the value of the slope of a tangent line for any point along that function. In this unit we will explore the many methods for calculating a derivative and the many interpretations of what a derivative means.

The derivative of a function f(x), is f’(x), which is equal to

f'(x) = showing limit as h appoaches zeroopening bracket
f(x + h) - f(x)
h
closing bracket

This definition is the basic principle of how calculus works, we will go on to expand on it and write it in simpler more elegant forms.

NOTE: Below are different notations you may see to represent the derivative of a function, in this case, The derivative of a function f(x) can be written as,

f’(x)

D(f) or D(f(x))

Using the Definition to Calculate a Derivative

In these examples we will calculate the derivative using the fundamental formula.

Estimate of Slope and Derivative with Definition

Below we have the graphed function y = f(x) = 3.

x-y graph showing function f(x)=3 - a straight horizonatl line

a) Estimate the slope at x = -1, 0, 1

b) Use the fundamental formula:

f'(x) = showing limit as h appoaches zeroopening bracket
f(x + h) - f(x)
h
closing bracket

to determine the derivative of the function y = f(x) = 3.

a) Looking at our graph, we can see we have a straight horizontal line, which you may have realised given the function f(x) = 3. When a function equals a constant we will always have a straight horizontal line with the slope always equal to 0, no matter what x-value we have.

So we can say that the slope at point x = -1, x = 0 and x = 1 will all be 0. We can also try subbing in those x-values into the function, f(-1) = 3, f(0) = 3, f(1) = 3. We are always given the same value for y (from the function).

b) Now we try using the fundamental formula to calculate the derivative that would give us the the formula for the slope at each point x.

f'(x) = showing limit as h appoaches zeroopening bracket
f(x + h) - f(x)
h
closing bracket

Lets start by writing what f(x + h) and f(x) are equal to,

We know that f(x) = 3,

For f(x + h), we would normally sub in (x + h) into our x slot in the function, but here there is no x slot, so,

f(x + h) = 3

Now subbing these values into

f'(x) = showing limit as h appoaches zeroopening bracket
f(x + h) - f(x)
h
closing bracket

We get,

f'(x) = showing limit as h appoaches zeroopening bracket
3 - 3
h
closing bracket
f'(x) = showing limit as h appoaches zeroopening bracket
0
h
closing bracket

= 0,

Therefore, the derivative of the function f(x) is

f’(x) = 0

Theorem on Derivatives on functions that equal constants

If f(x) = k, for any number k, then f’(x) = 0

Derivative Graphically with Definition

a) Use the fundamental formula:

f'(x) = showing limit as h appoaches zeroopening bracket
f(x + h) - f(x)
h
closing bracket

to calculate the derivative of y = f(x) = 4x2.

b) Use the derivative found in a) to calculate the slope of the tangent at the point (1, 4)

a) Calculating the derivative of y = f(x) = 4x2 using the formula, we first write what f(x + h) and f(x) are:

f(x + h) = 4(x + h)2 =

4(x2 + 2xh + h2) =

4x2 + 8xh + 4h2

f(x) = 4x2

Sub these values into,

f'(x) = showing limit as h appoaches zeroopening bracket
f(x + h) - f(x)
h
closing bracket
f'(x) = showing limit as h appoaches zeroopening bracket
4x2 + 8xh + 4h2 - 4x2
h
closing bracket
f'(x) = showing limit as h appoaches zeroopening bracket
8xh + 4h2
h
closing bracket

h cancelling out on the bottom and eliminating some of the h terms on the top,

f'(x) = showing limit as h appoaches zero(8x + 4h)

Now we have f’(x) equals the limit of h approaching zero, so

f’(x) = ( 8x + 4(0) )

f’(x) = 8x,

So this gives us the derivative of the function f(x) = 4x2, this formula of f’(x) = 8x will tell us the slope of the tangent to any point, x in the function f(x).

b) Now we need to find the slope of the point (1, 4) in the function f(x), we do this by subbing in the x-value from (1, 4) into f’(x),

f’(x) = 8x

f’(1) = 8(1)

f’(1) = 8

So, the slope of the tangent at the point (1, 4) is 8.

Definition - Tangent Line Formula

A function is differentiable at a point, x, if the function has a derivative at x.

The Tangent line Formula:

If a function is differentiable at a, the tangent line equation to the function at a point (a, f(a)) is

y = f(a) + f’(a)(x - a)

Derivative of Sine and Cosine

Definition - Derivative of Sine and Cosine

D(sin(x)) = cos(x)

D(cos(x)) = -sin(x)

Practicalities and Formulas for the Derivative

There are many ways of representing a derivative, and many terms that are commonly used in the world which are the same as a derivative. Below is a list of some of them.

Physics

Velocity - If the position of an object at time x is f(x) , then the velocity of the object at time x is f’(x). If the units for x are hours and f(x) is distance measured in miles, then the units for f '(x) = df dx are miles hour , miles per hour, this is velocity.

Acceleration - If the velocity of an object at time, x is f(x), then the acceleration of that object is represented by f’(x). If the velocity is in miles per hour, then the acceleration is in miles per hour per hour or miles / hour2.

Magnification - For a function f(x), the magnification of that function is f’(x).

Business

Marginal Cost - If the total cost of x is f(x) then the marginal cost is f’(x) at production level of x.

Marginal Profit - If the total profit from producing and selling a product x is f(x) then f’(x) is the marginal profit.

Derivative of a Power - The Power Rule

In Differentiation we will more often than not be dealing with variables that have exponents. The Power Rule is a very useful formula and method for differentiating these exponents.

Definition - Power Rule

The Power Rule - If n is a positive integer and x is any variable,

then the derivative of xn, (D(xn)) is n(x(n-1))

D(xn) = n(x(n-1))

So to derive a variable with an exponent, the power rule is to multiply the original variable by the value of the exponent, and then subtract the exponent of the variable by one. We will practice this method below, but one way to always think of it is “multiply down and subtract 1 from the power”.

Checkout More Power Rule Examples