Find the Derivative

Recall from algebra how to find the slope of a line: Given two points on a line,  and , the formula for the slope of the line is given by:

This equation works great if we are given a line, where the slope doesn’t change depending on what points we take. But what if we want to estimate the slope of a curve at a certain point? We can do this by picking two points very close to each other and finding the slope of the line between them. The closer the points we pick are to each other, the closer their slope will be to the slope at one of the points.

By modifying our formula slightly, we can keep track of how close the points are to each other. In order to estimate the slope of some function, $f(x)$ at a point $(x, f(x))$, we chose another point a small distance away from the first: $(x + h, f(x + h))$. Then the formula for the slope of this line becomes:

The smaller h gets, the closer the slope of our line approximates the slope of the curve:

This line, which crosses our curve at two points, is called a secant line. But while the slope of our line gets very close to the slope at the point, as long as h is greater than 0, it will never be exact. And we cannot set h equal to zero, since it is on the denominator of the fraction. So how can we find the slope exactly? We can take the limit as h approaches 0, and if the limit exists, that limit is the slope of a line that touches our curve only at the point $(x, f(x))$. This line is called a tangent line, and its slope is what we were looking for. We will call the slope of the tangent line the derivative of the curve at the point $(x, f(x))$, and represent the derivative by $f'(x)$ In other words,

There are a few other ways to notate the derivative:

Another way to think about derivative is rate of change. If the derivative is positive, we say our function is increasing; and if the derivative is negative, we say our function is decreasing. If the derivative is zero, then the function is neither increasing or decreasing.

Example: Using the limit definition, find the derivative of $f(x) = 3x^2 - 2x + 8$ at the point $x = 5$.

In order to solve this one, we’re going to have to use the formula for derivative:

Plugging 5 in for x gives us:

Next, let’s evaluate f(5+h) and f(5):

Notice how, after some simplification, the h cancelled out, allowing us to plug in to obtain the answer. For most simple functions, this will be the case.

You can also take a derivative without substituting a value for x. This gives you an equation to find the slope of the line at any point x; this equation is also called the derivative.

Example: Using the limit definition, find the derivative of $f(x) = x^2 + 2x - 5$. Then find $f'(0), f'(3), \text{ and } f'(1082)$. Just like last time, we’ll use the limit definition of the derivative:

However, this time we won’t plug in a value. Instead, we’ll substitute in the function and simplify without touching x:

Now that we’ve obtained a formula for $f'(x)$, we can easily evaluate the derivative at any point:

$f'(0) = 2(0) + 2 = 2$          $f'(3) = 2(3) + 2 = 8$          $f'(1082) = 2(1082) + 2 = 2166$

While the limit definition gives us the precise value of the derivative of a function, it can be time-consuming. Fortunately, just as we did in the second example above, you can easily use the limit definition to create a derivative formula for many functions. Below are a list of rules for the derivatives of many common functions:

There are other basic functions with derivatives, but these are the most common. Any function which has a derivative is called differentiable.

Example: Find the derivative of $f(x) =5^x$.

We consult the chart and see that $\frac{d}{dx}n^x = n^x \ln(x)$.  Therefore, $f'(x) =5^x \ln(5)$.

Example: Find the derivative of $f(x) = \sqrt{x}$.

Recall that $\sqrt{x} = x^{1/2}$.  Therefore, we can use the rule $\frac{d}{dx} x^n = nx^{n - 1}$, and we obtain $f'(x) = \frac{1}{2} x^{-1/2}$ This rule is often called the power rule.

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