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In this section we'll find derivatives of functions that pop up all the time. Unlike pop-up ads, they won't be blocked. Almost all derivative problems will use one of these functions. We'll look at constant functions, lines, power functions, exponential functions, and some logarithmic and trigonometric ones too.
The first kind of function is a constant function.
If f(x) = C for some constant C, then f ' (x) = 0.
Constant functions have a slope of zero. On a graph, a constant function is a straight vertical line. It doesn't matter if we pick f(x) = 3 or f(x) = -10, or f(x) = π, or any other constant, the result would be the same. Zero. We like.
We've seen lines before. The restroom at half-time, the blue things on notebook paper...and they were all over algebra. y = mx + b, anyone? Lines are a special case of polynomials. Once we master them, we'll be ready to add some power. As in the power of 2. Not rangers.
Lines have a constant slope. If we write our line as y = mx + b, the slope will be m. But wait, isn't there a relationship between slope and derivative?
Yeah, they're kinda the same thing. Since lines have a constant slope the derivative of any line will just be m.
To recap, if f(x) = mx + b is a line, then f ' (x) = m. The derivative will be constant, and equal to the slope of the line for every value of x.
A power function is any function of the form f(x) = xa, where a is any real number.
The following are all power functions:
The following are all power functions, written deceptively.The function
is a power function since it can be written as
f(x) = x1/2
f(x) = x.5.
is also a power function, since this can be written as
g(x) = x–6.
f(x) = xx
is not a power function, because the exponent is a variable instead of a constant.
We usually assume the exponent a isn't 0, because if a is 0 we find a power function. The function
f(x) = x0 = 1
is a constant function, and we already know how to deal with those.
Now that we've got an idea of what a power function is we can talk about their derivatives. Luckily, there's a handy rule we can use to find the derivative of any power function that we want.
Given a power function, f(x) = xa, the power rule tells us that
f '(x) = axa – 1
To find the derivative, just take the power, put in front and then subtract 1 from the power.
Using this rule, we can quickly find the derivative of any power function. The derivative of x2 is 2x, x1.5 is 1.5x0.5, and xπ is πxπ – 1.
No matter the power function, we can find its derivative.
A power function has a variable x in the base and a constant for the power. An exponential function has a constant for the base and a variable for the power:
f(x) = ax.
In order to make life easier (we do that sometimes) we assume a is not 0, 1, or negative. If a is 0 then our function is f(x) = 0x, which is undefined when x = 0 and is 0 everywhere else. Not too interesting.
If a is 1 then our function is f(x) = 1x = 1, which is a constant function, so also not too interesting. If a is negative, the function f(x) = ax is too weird to deal with: f will be negative some places, positive some places, and undefined at a lot of places (such as when x = 0.5.).
The following are exponential functions:
f(x) = 2x
f(x) = ex
f(x) = (0.5)x
f(x) = xx
is not an exponential function because it has a variable for the base and the power.
When we think of "exponential function," a good default function to think of is
f(x) = ex.
If we look at the estimates from the previous exercise, we're estimating that when f(x) = ex,
It turns out this is accurate. The derivative of the function f(x) = ex is
f'(x) = ex.
f(x) = ex is its own derivative.
If f(x) = ex then f ' (4) = e4.
Of course, there are exponential functions with other bases besides e. We'll give the derivative rule here, and the reasons after we talk about the chain rule.
If f(x) = ax, then f ' (x) = ax ln a.
If f(x) = 2x, then f ' (x) = 2x ln 2. To find the value of the derivative at a specific value of x, we plug that value in for x in the derivative function:
f ' (4) = 24 ln 2 = 16ln2.
Logarithmic functions come in different flavors and bases. There's vanilla, strawberry...sugar cone, dish...but we'll start with the most basic one:
f(x) = ln x.
This function is defined for x > 0, and looks like this:
Think about the slope/derivative of this function. First off, since the function f(x) = ln x is always increasing, its derivative is always positive. Also, since f(x) = ln x is only defined for x > 0, f ' will also only be defined for x > 0. The graph of f ' will be entirely in the first quadrant:
If we take x = a close to zero, then the slope of f at a will be very steep:
Therefore f ' (a) will be large:
As x = a gets closer to zero, f ' (a) will be even larger:
If we take x = a far from zero, then the slope of f at a will be shallow, f ' (a) will be close to zero:
As x = a gets farther from zero, f ' (a) will move closer to zero:
If we fill in this rather sketchy graph of f ', we find the graph of for x > 0:
Here's the rule for finding the derivative of the natural log function:
If f(x) = ln x, then
The graph is useful for remembering this rule. After we introduce the chain rule we'll see another way to find the derivative of ln x.
The trigonometric functions are the functions used often in trigonometry:
sin(x), cos(x), tan(x), csc(x), sec(x), and cot(x).
We'll find the derivatives of some of these now, and some we'll leave until we've learned more rules.
Start with f(x) = sin(x). While we could calculate the derivative of this function using the limit definition, we'll do what we did with the logarithmic function and use pictures.
The graph of f(x) = sin(x) looks like this:
Wherever the function hits a maximum or minimum value, it has a horizontal tangent line:
Wherever there's a horizontal tangent line, the derivative is zero:
Estimating f ' (0) = 1, f ' (π) = -1, and f '(2π) = 1 (Exercise: estimate these derivatives using tables.) we find the following rough graph of the derivative:
Since we already know the answer, we'll connect the dots.:
And ... da da da dum... the derivative of f(x) = sin(x) is
f ' (x) = cos(x).
Now look at f(x) = cos(x):
Again, we can find all the places the derivative is zero by finding all the maxima and minima of f:
Estimating with tables we find f ' (0) = 0, f ' (π/2) = -1, and f ' (3π/2) = -1 (Exercise: estimate these derivatives using tables):
Filling in the dots, we find
The derivative of f(x) = cos(x) is
f ' (x) = -sin(x).
Be Careful: To remember that the derivative of sin(x) is +cos(x), and the derivative of cos(x) is -sin(x), look at the graphs. The graph of sin(x) starts by increasing, so its derivative must be positive at first:
The graph of cos(x) starts by decreasing, so its derivative must start out negative:
The other trig functions can all be written as quotients involving the sine and/or cosine functions:
To find the derivatives of these functions, first we need to know how to find the derivatives of quotients.
Deriving inverse trig functions, such as arcsin(x) and arccos(x), requires knowledge of derivatives of inverse functions in general. This uses the chain rule, which we'll discuss later.