Derivative of a Function: Definition & Example

Instructor: Staci Corbett

Holding a Master of Business Administration, Staci has taught and tutored Mathematics for over 10 years.

There are two ways to define and many ways to find the derivative of a function. In this lesson, we will define the derivative using the instantaneous rate of change and provide examples.

Introduction: Slope of a curve

We all know how to find the slope of a straight line. It's been hammered into our heads since 7th grade. You simply divide the change in y by the change in x - or as our elementary school math teacher stated rise over run. This is commonly known as the rate of change. While this works really well for straight lines, what happens when we want to find the slope of a nonlinear function? Will the same approach work? The answer is no. The derivative, which is defined as the instantaneous rate of change or the slope at a specific point of a function, can help us overcome this challenge.

Slopes of linear equations are constant across the entire line. Intuitively, you may tell yourself that since we are considering a curve there won't be a constant slope for the entire function. That assumption would be correct! When attempting to find the slope of curves, we seek to find an equation that we can use to give us the slope of a line tangent to the curve at any given value of x. Using this equation gives us the instantaneous rate of change - or the slope at a specific point on the curve.

Definition and Formula

The equation that we find is known as the derivative of the function. As previously stated, the derivative is defined as the instantaneous rate of change, or slope, at a specific point of a function. It gives you the exact slope at a specific point along the curve. The derivative is denoted by (dy/dx), which simply stands for the derivative of y with respect to x.

Finding the derivative or slope of a curve at a specific point is an application of the topic of limits that you've learned previously. To find the derivative, use the following formula:

derivative formula

Example of determining instantaneous rate of change using the derivative

One of the most beneficial features of the derivative is it's ability to ease the burden of determining instantaneous rate of change. The below example illustrates how the derivative function can be used to easily calculate instantaneous rate of change at a specific point on a curve.

Find dy/dx if y=4x^2 + 6x

dy/dx = lim h->0 ((4(x+h)^2 + 6(x+h)) -(4x^2 + 6x) )/ h

dy/dx = lim h->0 (4(x^2 +2xh+h^2) + 6x +6h) -4x^2-6x/h

dy/dx = lim h->0 (4x^2 + 8xh + 4h^2 + 6x +6h - 4x^2 -6x)/h

dy/dx =lim h->0 (8xh + 4h^2 +6)/h

dy/dx = lim h->0 h(8x +4h + 6)/h

dy/dx = lim h->0 8x + 4h + 6

dy/dx = 8x +6

What does dy/dx = 8x + 6 tell us?

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