It is tedious to compute a limit every time we need to know the derivative of a function. Fortunately, we can develop a small collection of examples and rules that allow us to quickly compute the derivative of almost any function we are likely to encounter. We will start simply and build up to more complicated examples.

First, we introduce a different notation for the derivative which may be more convenient at times.

1 The constant rule

The simplest examples of functions, hence the best place to start our investigation, are constant functions. Recall that derivatives measure the rate of change of a function at a given point. This means the derivative of a constant function is zero. Here are some ways to think about this situation.

• The constant function plots a horizontal line—so the slope of the tangent line is \(0\).
• If \(s(t)\) represents the position of an object with respect to time and \(s(t)\) is constant, then the object is not moving, so its velocity is zero. Hence \(\frac {d}{dt} s(t) = 0\).
• If \(v(t)\) represents the velocity of an object with respect to time and \(v(t)\) is constant, then the object’s acceleration is zero. Hence \(\frac {d}{dt} v(t) = 0\).

The examples above lead us to our next theorem. To gain intuition, you should compute the derivative of \(f(x) = 6\) using the limit definition of the derivative.

2 The power rule

Next, let’s examine derivatives of powers of a single variable. To gain intuition, you should compute the derivative of \(f(x) = x^3\) using the limit definition of the derivative. Before computing this derivative, we should recall the Binomial Theorem.

The Binomial Theorem tells us the pattern of the coefficients for the expanded form of

which will help us prove our next derivative rule.

The power rule For any real number \(n\),

\[ \frac {d}{dx} x^n = n x^{n-1}. \]

At this point we will only prove this theorem for values of \(n\) which are positive integers. Later we will give the complete explanation. From the limit definition of the derivative, write with me

\[ \frac {d}{dx} x^n = \lim _{ h\to 0} \frac {\left (\answer [given]{x+ h}\right )^n-x^n}{h}. \]

Expand the term \((x+h)^n\):

Note that we are using the Binomial Theorem to write \(\binom {n}{k}\) for the coefficients. Canceling the terms \(x^n\) and \(-x^n\), and noting \(\binom {n}{1}= n\), write with me

since every term but the first has a factor of \(h\).

Therefore,

\[ \frac {d}{dx} x^n = nx^{n-1}. \]

Let’s consider several examples. We begin with something basic.

Sometimes, it is not as obvious that one should apply the power rule.

The power rule also applies to radicals once we rewrite them as exponents.

3 The sum rule

We want to be able to take derivatives of functions “one piece at a time.” The sum rule allows us to do exactly this. The sum rule says that we can add the rates of change of two functions to obtain the rate of change of the sum of both functions. For example, viewing the derivative as the velocity of an object, the sum rule states that to find the velocity of a person walking on a moving bus, we add the velocity of the bus and the velocity of the walking person.

The sum rule If \(f(x)\) and \(g(x)\) are differentiable and \(c\) is a constant, then

(a)

\(\frac {d}{dx} \big ( f(x) + g(x)\big ) = f'(x) + g'(x)\),

(b)

\(\frac {d}{dx} \big ( f(x) - g(x)\big ) = f'(x) - g'(x)\),

(c)

\(\frac {d}{dx} \big (c\cdot f(x)\big ) = c\cdot f'(x)\).

We will only prove part (a) above; the rest are similar. Write with me

We now have the tools to work some more complicated examples.